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Year 7 Western Australia Curriculum

© 2023 Mathspace Pty Ltd Copyright Notice This Work is copyright. All rights are reserved. Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL). Reproduction and communication for other purposes Except as permitted under the Act (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher. For permission to use material from this text or product, please email hello@mathspace.com.au For our full digital offering, visit mathspace.co Mathspace Pty Ltd Level 11, 477 Pitt St Sydney 2000 School Curriculum and Standards Authority. Mathematics Scope and Sequence of the mandated curriculum content: Pre-primary – Year 10 | Revised Curriculum for familiarisation in 2025. The School Curriculum and Standards Authority does not endorse this publication or product. Title: Mathspace WA Curriculum - Year 7: 2026 ISBN: 978-1-963022-77-3 Editors and lead authors: Erin Gallagher, Ronnel Jeppsen, Julia Reid, Sarah Butterworth, Anjelica Sieger, Jennifer George, Stephen George, Dot Yung Further reviewers: Melissa Archey, Warren Bardsley, Lianna Beeching, Kathryn Bennett, Benjamin Bui, Eric Chan, Cory Duker, Bahija Elmir, Wayne Gillie, Jeremy Hunter, Geoff Jamieson, Shahad Odeh, Natasha Pollard, Helen Soar, Patricia Toutzaris, Paul Unkles, Jie Yang Writing and development team: Adam Humphries, Neil Lopez, Ma. Lourdes Ona, Bernadette Barillo, Jelly Candido, Sheramie Castillo, Dhave Fernandez, Abegail Fontanilla, Regine Jara, Albert Jara, Shaira Llanita, Lesley Lopez, Kerby Mendoza, Mikaela Nicolas, Rolly Apdo, Valerie Baja, Marielle Belen, Michaela Belen, Christine Bibon, Allen Calabia, Adrian Doctolero, Dharent Fernandez, Janine Francisco, Rose Francisco, Rona Josue, Crystel Lontoc, Mae Lucid, Grace Manzano, Julie Manzano, Monisa Tacud, Paul Platero Images and Design team: Alex Duckmanton, Chastine Marquez, Keith Gimeno

Contents 1

Whole numbers

1.01

Addition and subtraction

1.02

Multiplication

1.03

Division

1.04

Order of operations with whole numbers

1.05

Check reasonableness of calculations

Investigation: How long does it take to count to a million?

1.06

Number calculations with technology

1.07

Write mathematical statements

Modelling Investigation: Charity fundraising

Chapter 1 review

Powers, factors and multiples

2.01

Number operation properties

2.02

Square and cube numbers

2.03

Square and cube roots

2.04

Powers of whole numbers

2.05

Powers of ten

2.06

Factors and multiples

Investigation: The sieve of Eratosthenes 2.07

Prime factorisations

Investigation: Celtic knots 2.08

HCF and LCM

Chapter 2 review

Integers 100 3.01

Integers and the number line

102

3.02

Compare integers

109

3.03

Add and subtract integers

114

Chapter 3 review

122

Investigation: Life-size number line

Contents mathspace.co

iii

Fractions 124 4.01

Identify, order and compare fractions

126

4.02

Negative fractions

134

4.03

Equivalent and simplified fractions

140

Investigation: Fraction flowers 4.04

Add and subtract fractions

150

4.05

Fractions of a quantity

158

4.06

Multiply fractions

164

4.07

Divide fractions

170

4.08

Order of operations with fractions

177

Chapter 4 review

182

Investigation: Egyptian fractions

Decimals 186 5.01

Decimals and place value

188

5.02

Add and subtract decimals

194

5.03

Convert between fractions and decimals

200

5.04

Multiply decimals

209

5.05

Divide decimals

216

5.06

Round decimals

221

Investigation: Account statements Chapter 5 review

Percentages

229

232

6.01

Fractions, decimals and percentages

234

6.02

Compare percentages

243

6.03

Express a quantity as a percentage

247

6.04

Percentages of a quantity

252

6.05

Percentages with technology

259

Chapter 6 review

265

Ratios 268 7.01

Identify ratios

270

7.02

Equivalent and simplified ratios

275

7.03

Find part or whole measures

282

Chapter 7 review

290

Mathspace WA Curriculum – Year 7 mathspace.co

Algebra 292 8.01

Introduction to variables

294

8.02

Build algebraic expressions

299

8.03

Substitution

305

8.04

Collect like terms

311

Chapter 8 review

317

Equations 320 9.01

Introduction to equations

322

9.02

Write rules for relationships

329

9.03

One-step equations

338

9.04

Two-step equations

343

Chapter 9 review

353

Cartesian plane

356

10.01

The Cartesian plane

358

10.02

Directed numbers on the Cartesian plane

366

10.03

Graphs of linear relationships

376

Chapter 10 review

386

Angles, lines, and shapes

390

11.01

Geometry notation and language

392

11.02

Measure angles

401

11.03

Parallel lines and transversals

410

11.04

Triangles

422

11.05

Quadrilaterals

432

Investigation: Sort shapes with algorithms Chapter 11 review

442

Contents mathspace.co

Length and area

448

12.01

Units of length

450

12.02

Perimeter

456

12.03

Perimeter of composite shapes

466

12.04

Units of area

472

12.05

Area of rectangles and squares

480

12.06

Area of triangles

489

12.07

Area of composite shapes

496

Modelling investigation: Solar panels

506

Chapter 12 review

509

Solids and volume

512

13.01

Nets of solids

514

13.02

Visualise solids

519

13.03

Volume using unit cubes

538

13.04

Units of volume

542

13.05

Volume of right prisms

547

13.06

Attributes in measurement formulas

556

Chapter 13 review

570

Transformations 574 14.01

Transformation of shapes

576

14.02

Translations

582

14.03

Reflections

591

14.04

Rotations

599

14.05

Multiple transformations

610

14.06

Line and rotational symmetry

622

Chapter 14 review

631

Investigation: Tessellations

Mathspace WA Curriculum – Year 7 mathspace.co

Probability 636 15.01

Language of probability

638

15.02

Probability as a number

646

15.03

Theoretical probability

652

15.04

Relative frequency

658

15.05

Venn Diagrams

666

Modelling investigation: Recycling probability

678

Investigation: Simulations Chapter 15 review

Statistics

680

684

16.01

Types of data

686

16.02

Frequency tables

691

16.03

Summarise data

697

16.04

Outliers

706

16.05

Dot plots and column graphs

713

16.06

Stem and leaf plots

727

16.07

Pie charts

735

16.08

Line graphs

743

Investigation: Statistical investigations Chapter 16 review

753

Time 758 17.01

Elapsed time

760

17.02

12 and 24-hour time

765

17.03

Australian time zones

769

Chapter 17 review

775

Answers

778

Contents mathspace.co

vii

Big ideas Arithmetic skills, including order of operations, estimation, technology and symbolic expressions, are essential tools for problem - solving and building a strong foundation for advanced mathematical concepts.

Curriculum Outcomes Analyse, represent and interpret problems in real-world situations involving whole numbers, positive fractions, decimals and percentages, addition and subtraction of integers, numbers in index form, linear equations with up to two operations, simple number patterns and/or transactional money statements

1 Whole numbers Chapter outline 1.01 Addition and subtraction 1.02 Multiplication 1.03 Division 1.04 Order of operations with whole numbers 1.05 Check reasonableness of calculations Investigation: How long does it take to count to a million? 1.06 Number calculations with technology 1.07 Write mathematical statements Modelling Investigation: Charity fundraising Chapter 1 review

5 10 16 20 25 29 33 39 41

Whole numbers help construction workers calculate the exact materials needed for their projects!

“The beauty of mathematics only shows itself to more patient followers.” Maryam Mirzakhani

1.01 Addition and subtraction After this lesson, you will be able to… • add multi-digit numbers. • subtract multi-digit numbers.

Addition The most basic operation is addition, which uses the + symbol. The expression 5 + 3 means “3 more than 5”, which is the same as the number 8. Think about it as moving from left to right along a number line: Starting at 5 and taking 3 steps right lands on 8.

+3 0

2 3 4 5 6 7 8 9 10 5+3=8

One strategy is to use regrouping, where we split each number up into convenient pieces. We can then add the numbers together in an order that we ﬁnd easiest. Another strategy is to use the vertical algorithm, where we align our numbers using place value. Let’s look at an example.

Example 1 Find 271 + 95.

Create a strategy Use a vertical algorithm.

Apply the idea Write the results in a vertical algorithm and add the digits from the units column to the tens column: 1

2 7 1 9 5 3 6 6

In this case, when we added the values in the tens column, we regrouped into the hundreds column, giving us a ﬁnal answer of 366. So, 271 + 95 = 366.

1.01 Addition and subtraction mathspace.co

Idea summary We can add two or more numbers together in a single place value table. If any column adds up to more than 10, we need to carry the number of tens into the next column. We can use regrouping to add large numbers, where we split each number up into convenient pieces such as thousands, hundreds, tens and ones.

Subtraction If we want to separate quantities, we use subtraction. Subtraction uses the − symbol and is the opposite of addition. The expression 5 − 3 means a number “3 less than 5”, which is the same as the number 2. Think about it as moving from right to left along a number line: −3 0

Starting at 5 and taking 3 steps left lands on 2.

2 3 4 5 6 7 8 9 10 5−3=2

A subtraction equation is always related to an addition equation, like this: 13 − 7 = 6 is related to 6 + 7 = 13 is related to 13 − 6 = 7. We can perform subtraction using the vertical algorithm as well. Write the ﬁrst number above the second and work from right to left, subtracting the bottom number from the top number. If the bottom number is larger than the top number in any column, add 10 to the top number and take 1 away from the top number in the next column. Another strategy is to use regrouping, where we split each number up into convenient pieces. We can then subtract the numbers from each other in an order that we ﬁnd easiest.

Example 2 Evaluate 2221 − 1391.

Create a strategy Use a vertical algorithm.

Apply the idea 1. Write the subtraction in a vertical algorithm. -

Mathspace WA Curriculum – Year 7 mathspace.co

2 2 2 1 3 9

1 1

2. Begin with the ones column. Subtract the numbers, 1 − 1 = 0. -

2 2 2 1 3 9

1 1 0

3. Next, we move on to the tens column. Since we don’t have enough tens to subtract 9 from 2, we need to borrow 1 hundred from the hundreds place. 2 1 12 - 1 3 9

1 1 0

4. We can subtract 9 from 12 in the tens column and 2 hundreds becomes 1 hundred. 2 1 12 1 - 1 3 9 1 3 0 5. Next, we move on to the hundreds column. Since we don’t have enough hundreds to subtract 3 from 1, we need to borrow 1 thousand from the thousands place. -

1 11 12 1 1 3 9 1 3 0

6. Now we can subtract 3 from 11 in the hundreds column and 2 thousands become 1 thousand. Then subtract the numbers in the thousands column, 1 − 1 = 0. -

1 11 12 1 1 3 9 1 8 3 0

So, 2221 − 1391 = 830.

Example 3 Evaluate 968 − 353.

Create a strategy Split each digit of 968 and 353 based on its place value.

Apply the idea Let’s try splitting 968 into 900, 60 and 8 and 353 into 300, 50 and 3. Subtract 300, 50 and 3 from 900 + 60 + 8

Subtract 300 from 900

Subtract 50 from 60

Subtract 3 from 8

Add the numbers

1.01 Addition and subtraction mathspace.co

Idea summary We can perform subtraction using a place value table. Write the first number above the second and work from right to left, subtracting the bottom number from the top number. We can use regrouping to subtract large numbers, where we split each number up into convenient pieces such as thousands, hundreds, tens and ones.

1.01 Practice questions Understanding 1

More than

Ex 2, 3

Sum

Fewer

Increase

Difference

Less than

Total

Minus

Plus

Decrease

Combined

Reduce

93 fewer than 2931

Evaluate: 321 more than 8537

472 reduced by 386

the difference between 554 and 26

the sum of 781 and 933

the total of 88 and 662

656 decreased by 174

643 increased by 842

Fluency 3

Trailblazer 1, 2e-h

Ex 1

Adventurer 1, 2bdfh

Decide if these words represent addition or subtraction: a

Explorer 1, 2aceg

3a-d, 4a-di

3e-h, 4e-h

3i-l, 4e-h

Calculate: a

77 + 22

810 + 180

672 + 119

7017 + 2906

6347 + 332

63 423 + 1272

55 236 + 4004

882 + 565 + 53

676 + 235 + 46

536 + 492 + 698

50 + 8261 + 938 + 4394

1635 + 84 + 249 + 9874

Calculate: a

99 999 − 300

67 773 − 3003

66 747 − 25 332

3650 − 43

2519 − 146

99 447 − 7718

83 849 − 24 187

56 253 − 47 785

953 – 121

Mathspace WA Curriculum – Year 7 mathspace.co

Reasoning

5–6

6–8

Explain why another way to calculate 18 + 12 is to calculate 20 + 10.

Explain why another way to calculate 342 + 35 is to calculate 300 + 77.

Determine whether these statements are true or false:

Regrouping is a method of breaking down small numbers into larger pieces to make addition and subtraction easier.

When adding large numbers, it is important to keep track of the values that you carry over to the next column.

We can use regrouping to add large numbers, where we split each number up into convenient pieces such as thousands, hundreds, tens and ones.

When adding large numbers, it is not necessary to line up the columns correctly.

We can perform subtraction using a place value table. We write the ﬁrst number above the second and work from right to left, subtracting the bottom number from the top number.

The difference between two whole numbers is always a whole number.

Neil is adding two 3-digit numbers: 5 5 A + 2 B 2 C 6 1 Find the missing values of A, B and C.

Problem-solving 9, 11 9

9–12

Four friends are picking fruit in an orchard. Kathleen picks 27 apples, Xavier picks 25 peaches, Ryan picks 13 pears and Elizabeth picks 15 plums. How many pieces of fruit were picked in total?

There are 232 different species of fish and other wildlife at an aquarium. The aquarium has advertised the addition of 92 new species of fish, 8 new species of crustaceans and 9 new species of sharks. How many species of fish and wildlife will the aquarium be home to?

It took Gwen 137 seconds to finish a race. John finished the race 3 seconds quicker than Gwen and Susana finished the race 4 seconds quicker than John. How many seconds did it take Susana to finish the race?

Georgia has ordered 388 large bricks for the purpose of building a brick shed. She uses 222 bricks for the two side walls, 82 bricks for the back wall and 54 bricks for the front wall, leaving space for a door. How many bricks will there be left over?

1.01 Addition and subtraction mathspace.co

1.02

Multiplication After this lesson, you will be able to… • multiply multi-digit numbers. • use different methods to multiply small and large numbers.

Multiplication with area models The expression 5 × 3 results in the same total as repeatedly adding 5 three times: 5 + 5 + 5 = 15. The expression 5 × 3 tells us in a more eﬃcient way that we have 3 groups of 5. We can connect multiplication and area by modelling 5 × 3 in a grid of squares. We can see that we need 15 small squares to cover a rectangle that is 5 units long and 3 units high:

1 1

5 × 3 = 15

11 12 13 14 15 5

Just like with addition, we can use the commutative law to change the order in which numbers appear in the expression, but that doesn’t change the result. Here we have a rectangle that is 3 units long and 5 units high and we can see it uses the same number of small squares to cover it: The answer to a multiplication is called the product. The product of 3 and 5 is 15.

3 × 5 = 15

Interactive exploration

Discover this concept in action online

mathspace.co

We can use an area model to visualise multiplying larger numbers. We can draw a large rectangle that is divided into smaller rectangles using place values, like ones, tens, hundreds etc. Looking at the area model we can also see that 5 × 3 has the same value as 3 × 5. This is called the commutative property, it means that multiplications can be done in any order.

Mathspace WA Curriculum – Year 7 mathspace.co

Example 1 Use the area model to find 76 × 68. a Fill in the areas of each rectangle. ×

60 8

Create a strategy

Apply the idea

Multiply the number at the top of the column by the number on the left of the row.

Area 1 = 60 × 70 = 4200 Area 2 = 60 × 6 = 360 Area 3 = 8 × 70 = 560 Area 4 = 8 × 6 = 48

Multiply 6 tens by 7 tens Evaluate Multiply 6 tens by 6 Evaluate Multiply 8 by 7 tens Evaluate Multiply 8 by 6 Evaluate

Filling in the areas of each rectangle, we have × 60 8

70 4200 560

6 360 48

b What is the total area of all four rectangles?

Create a strategy Add all the areas calculated in part (a).

Apply the idea Total area = 4200 + 560 + 360 + 48 = 5168

Add all the values Evaluate

c So what is 76 × 68?

Create a strategy

Apply the idea

Use the answer calculated on part (b).

76 × 68 = 5168

1.02 Multiplication mathspace.co

Idea summary Grid squares and area models use place value to help us visualise the result of multiplication.

Multiplication using an algorithm Once you understand how place value and area are connected, we can begin to use a place value table to multiply large numbers together more efficiently. To multiply numbers, we can align the numbers using a vertical algorithm. The same process can be used regardless of how many digits we have, we just have to repeat the steps a few more times.

Example 2 Find 243 × 15.

Create a strategy Use the standard algorithm method for multiplication.

Apply the idea 1. Write the product in a vertical algorithm: ×

2 4 3 1 5

2. Start from the far right. Multiply to get 5 × 3 = 15. So we write the 5 in the ones place and carry the 1 to the tens place: 2 14 3 × 1 5 5 3. Move left and multiply to get 5 × 4 = 20. Then we add the 1 to get 21. So we write the 1 in the tens place and carry the 2 to the hundreds place: 2

2 14 3 × 1 5 1 5

Mathspace WA Curriculum – Year 7 mathspace.co

4. Move left and multiply to get 5 × 2 = 10. Then we add the 2 to get 12. So we write 2 in the hundreds place and 1 in the thousands place: 2

2 14 3 1 5 1 2 1 5

5. Since we are multiplying by a number in the tens place, we will place a 0 in the ones place. Move left and multiply to get 1 × 3 = 3 so we put the 3 in the tens place. 2

2 14 1 1 2 1 3

3 5 5 0

6. Move left and multiply to get 1 × 4 = 4 so we put the 4 in the hundreds place. 2

2 14 1 1 2 1 4 3

3 5 5 0

7. Move left and multiply to get 1 × 2 = 2 so we put the 2 in the thousands place. 2

2 14 × 1 1 2 1 2 4 3

3 5 5 0

3 5 5 0 5

8. Add our two answers to get the ﬁnal answer: 2 14 × 1 1 2 1 2 4 3 3 6 4 This means we have 243 × 15 = 3645

Idea summary To multiply larger numbers, it can be more eﬃcient to use a vertical algorithm. Using a place value table can help us keep the place values aligned, so that we can add the results of our multiplication to ﬁnd the total result.

1.02 Multiplication mathspace.co

1.02 Practice questions Understanding

Explorer 1–2

Adventurer

Trailblazer

1–2

Explain using a diagram of a rectangle why 5 × 3 is the same as 3 × 5.

Tom made an area model to calculate 12 × 8. How does Tom find the answer to 12 x 8 from this? 8

Fluency Ex 1

3–4, 7a-h, 8a-d, 9a-d, 10a-d

3, 5, 7i-l, 8e-h, 9e-h, 10c-f

5–6, 7i-l, 8i-l, 9e-l, 10cgh

Let’s calculate 50 × 83 using an area model.

Find the areas of each of the smaller rectangles.

Using the smaller areas found in part (a), what is 50 × 83?

Let’s calculate 6 × 608 using an area model. a

Find the areas of each of the smaller rectangles.

Using the smaller areas found in part (a), what is 6 × 608?

Find the areas of each of the smaller rectangles.

Using the smaller areas found in part (a), what is 83 × 660?

600

Let’s calculate 83 × 660 using an area model. a

80 3

Let’s calculate 87 × 974 using an area model.

900

Find the areas of each of the smaller rectangles.

Using the smaller areas found in part (a), what is 87 × 974?

80 7

Calculate: a

2×3

7×1

9×4

1×8

5×5

6×2

3×7

3 × 10

2 × 15

11 × 7

12 × 3

5 × 12

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 2

Calculate: a

6 × 30

45 × 6

87 × 3

39 × 9

432 × 9

789 × 3

567 × 4

876 × 5

5555 × 3

1234 × 9

1298 × 2

3569 × 3

Calculate: a

4000 × 70

500 × 60

10 000 × 120

700 × 30

234 × 17

678 × 23

842 × 15

741 × 14

1234 × 19

3569 × 13

5873 × 42

9326 × 51

Calculate: a

2×5×9

3×2×8

10 × 7 × 9

12 × 2 × 2

4 × 2 × 15

20 × 10 × 6

12 × 5 × 4

15 × 15 × 5

Reasoning

11–12

11, 12ac

Explain why 5 × 6 is the same as 10 × 3.

Write these expressions as a sum of two products. Do not evaluate the product. a

8 × 75

11 × 92

Problem-solving 13a, 14–17 13

82 × 64

13ab, 14–18

475 × 73

13b-d, 14–20

Find two numbers that match these criteria. a

Product of 6 and sum of 5

Product of 24 and difference of 10

Product of 36 and sum of 13

Product of 1296 and difference of 0

Tristan works 8 hours a day, 6 days a week; and Sheila works 9 hours a day, 5 days a week. a

How many hours does Sheila work each week?

Who works more hours each week?

A rock climber descends from the top of a cliff face to the ground in 5 stages, dropping 106 metres each time. What is the height of the cliff face?

Maximilian builds handmade designer dining tables, each table sells for $4865. He has just received a new order for 6 tables. How much are the total sales if all 6 tables are sold?

A showroom has 155 electric bikes. Each electric bike cost $3500. What is the total cost of all the electric bikes?

A tyre company can produce 2512 tyres a day. How many tyres can they produce in 360 days?

1.02 Multiplication mathspace.co

A group of countries have decided to donate $3 508 060 every year towards improving freshwater availability. How much money will the group donate over 16 years?

A carton holds 36 packets of biscuits. Each packet has 12 biscuits. How many biscuits can be packed in 55 cartons?

1.03

Division After this lesson, you will be able to… • divide multi-digit numbers. • use different methods to divide small and large numbers.

Division Division uses the ÷ symbol and is essentially the opposite of multiplication. The expression 15 ÷ 3 means “15 split into 3 equal groups”, which is the same as the number 5.

Interactive exploration

Discover this concept in action online

mathspace.co

A division equation is (almost) always related to a multiplication equation: 20 ÷ 5 = 4 is related to 4 × 5 = 20 which is also related to 20 ÷ 4 = 5. When dividing larger numbers, we will use a procedure called short division. We write the number we are dividing by (the divisor) on the left and the number we are dividing (the dividend) on the right and draw a division bracket between them. We will be writing our result (called the quotient) above the bracket.

With short division, we are still dividing the dividend into equal groups of the divisor. However, we consider each place value of the dividend separately, starting with the largest place values.

Mathspace WA Curriculum – Year 7 mathspace.co

Example 1 Evaluate 1616 ÷ 4.

Create a strategy Use short division.

Apply the idea Form the short division tableau with 1616 as the dividend and 4 as the divisor. We cannot divide 1 by 4, so we carry the 1 across to the next number along to form 16. 16 ÷ 4 is 4, so we put the 4 in the hundreds place.

Again, we cannot divide 1 by 4, so we carry the 1 across to the next number along to form 16. We put a 0 in the tens place.

16 ÷ 4 is 4, so we put the 4 in the ones place. Since there is no remainder at this step, we conclude that 1616 ÷ 4 = 404

Example 2 Find the quotient and remainder of 1890 ÷ 8.

Create a strategy Use short division.

Apply the idea Form the short division tableau with 1890 as the dividend and 8 as the divisor. We cannot divide 1 by 8. so we carry the 1 across to the next number along to form 18.

1.03 Division mathspace.co

18 ÷ 8 is 2 remainder 2, so we put the 2 in the hundreds place. We then carry the remainder 2 across to the next number along to form 29. 29 ÷ 8 is 3 remainder 5, so we put the 3 in the tens place. We then carry the remainder 5 across to the next number along to form 50. 50 ÷ 8 is 6 remainder 2, so we put the 6 in the ones place, then the remainder 2. We conclude that 1890 ÷ 8 = 236 r. 2

Idea summary When dividing larger numbers, we can use short division.

We write the number we are dividing by (the divisor) on the left and the number we are dividing (the dividend) on the right. We will be writing our result (called the quotient) above the bracket. If a number cannot be divided evenly, then we have a remainder and sometimes that remainder is further evaluated or estimated showing a decimal answer.

1.03 Practice questions Understanding 1

Explorer 1–2

Adventurer 1–2

Write an expression for each of these statements: a

25 divided by 5

The quotient of 24 and 8

49 split into 7 groups

Determine whether these statements are true or false: a

5 ÷ 10 is the same as 10 ÷ 5.

Any value divided by itself gives an answer of 1.

Any number divided by 0 is 0.

If a multiple of 10 is divided by 10 there will be no remainder.

Mathspace WA Curriculum – Year 7 mathspace.co

Trailblazer 1–2

Fluency Ex 1

Ex 2

3c-i, 4d-i, 5d-i, 6d-i

3f-l, 4g-l, 5g-l, 6g-l

Calculate: a

72 ÷ 8

108 ÷ 6

270 ÷ 9

360 ÷ 4

546 ÷ 6

828 ÷ 3

1300 ÷ 5

2056 ÷ 8

2996 ÷ 4

4786 ÷ 2

6349 ÷ 7

9207 ÷ 9

Calculate: a

480 ÷ 12

620 ÷ 10

888 ÷ 12

1800 ÷ 15

2655 ÷ 59

3872 ÷ 88

7850 ÷ 25

8372 ÷ 28

22 644 ÷ 34

31 512 ÷ 52

36 300 ÷ 60

39 000 ÷ 65

5555 ÷ 6

Find the quotient and the remainder for these divisions: a

3, 4a-ei, 5a-ei, 6a-ei

1242 ÷ 7

3126 ÷ 4

4108 ÷ 5

6125 ÷ 9

7134 ÷ 7

7318 ÷ 8

9312 ÷ 5

12 314 ÷ 4

23 968 ÷ 5

29 876 ÷ 8

31 597 ÷ 3

Find the quotient and the remainder for these divisions: a

435 ÷ 18

539 ÷ 73

728 ÷ 55

819 ÷ 37

3126 ÷ 29

4138 ÷ 39

6346 ÷ 22

9721 ÷ 18

10 998 ÷ 22

15 206 ÷ 31

21 491 ÷ 35

37 311 ÷ 29

Reasoning

7–9

12 ÷ 3 is not equal to 3 ÷ 12. If the two numbers are x and y, when is x ÷ y equal to y ÷ x?

Why is it important to avoid making the divisor equal to zero?

Rewrite 28 ÷ 3 in the form of divisor × quotient + remainder. Explain your thinking.

Problem-solving 10–13

10, 12–14

13–17

What is the smallest number greater than 30 that leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 7? Explain your answer.

Sandy baked 320 biscuits and divided them equally into 20 packets. How many biscuits did Sandy put in each packet?

Glenn needs 8 oranges to make a glass of orange juice. If Glenn has 400 oranges, how many glasses of orange juice can he make?

Andrea wants to put her 425 books in boxes. If each box can hold 25 books, how many boxes does she need for her books?

A school is able to gather 2068 canned items from a donation drive. Each of the 47 recipients will receive the same number of canned items. How many items does each recipient receive? 1.03 Division mathspace.co

A stadium has 4056 seats with 52 seats per row. How many rows of seats does the stadium have?

A rectangular garden measures 80 metres by 60 metres and you need to place a plant every 2 metres around the perimeter of the garden. How many plants will you need in total?

Fill in the blanks to make these statements correct. a d g

47⬚ ÷ 17 = ⬚⬚

65⬚ ÷ 21 = ⬚⬚

9⬚4 ÷ 21 = ⬚⬚

4⬚4 ÷ ⬚⬚ = 14

81⬚ ÷ ⬚⬚ = 24

999 ÷ 3⬚ = 2⬚

c f

5⬚0 ÷ 14 = ⬚⬚

83⬚ ÷ ⬚⬚ = 27

1.04 Order of operations with whole numbers After this lesson, you will be able to… • perform arithmetic operations using the order of operations.

Order of operations with whole numbers By following the order of operations, we can make sure we get the correct answer every time. It’s like following a recipe when we’re cooking — we need to add the ingredients in the right order to make sure our dish turns out just right. The order is as follows: • Evaluate operations inside grouping symbols such as brackets. • Evaluate multiplication or division going from left to right. • Evaluate addition or subtraction going from left to right.

Example 1 Given the expression 6 + 16 ÷ 4: a Which operation should we perform ﬁrst? A Add 6 and 16

Divide 16 by 4

Create a strategy

Apply the idea

Division comes before addition.

We should divide 16 by 4 ﬁrst before performing addition. So the correct answer is option B.

Mathspace WA Curriculum – Year 7 mathspace.co

b Evaluate 6 + 16 ÷ 4.

Create a strategy

Apply the idea 6 + 16 ÷ 4 = 6 + 4 Divide 16 by 4

Perform division ﬁrst before addition.

= 10

Evaluate the addition

Example 2 Evaluate: a (3 + 5) × 8

Create a strategy First, evaluate the addition inside the brackets, then multiply the result by 8.

Apply the idea (3 + 5) × 8 = 8 × 8 = 64

Evaluate the addition inside the brackets Evaluate the multiplication

b [(4 + 6) × 8] ÷ 5

Create a strategy First, evaluate the addition inside the round brackets, then evaluate the multiplication inside the square brackets and finally perform the division by 5.

Apply the idea [(4 + 6) × 8] ÷ 5 = [10 × 8] ÷ 5

Evaluate the addition inside the round brackets

= 80 ÷ 5

Evaluate the multiplication inside the square brackets

= 16

Evaluate the division

Example 3 Given the expression 4 + 18 − 5 × 2: a Evaluate the expression.

Create a strategy Evaluate the expression following the correct order of operations.

1.04 Order of operations with whole numbers mathspace.co

Apply the idea 4 + 18 − 5 × 2 = 4 + 18 − 10

Evaluate the multiplication

= 22 – 10

Evaluate the addition

= 12

Evaluate the subtraction

b Without changing the order of the numbers and operations, write the expression which would evaluate to 30.

Create a strategy We can use brackets to change the order the operations are done.

Apply the idea Put 18 − 5 inside the brackets so that subtraction will be evaluated ﬁrst followed by multiplication then addition. 4 + (18 − 5) × 2 = 4 + 13 × 2

Evaluate the subtraction inside the brackets

= 4 + 26

Evaluate the multiplication

= 30

Evaluate the addition

So the expression is 4 + (18 − 5) × 2. c Without changing the order of the numbers and operations, write the expression which would evaluate to 34.

Create a strategy We can use brackets to change the order the operations are done.

Apply the idea Put 4 + 18 − 5 inside the brackets so that addition and subtraction will be evaluated ﬁrst followed by multiplication. (4 + 18 − 5) × 2 = (22 − 5) × 2

Evaluate the addition inside the brackets

= 17 × 2

Evaluate the subtraction inside the brackets

= 34

Evaluate the multiplication

So the expression is (4 + 18 − 5) × 2.

Idea summary The order of operation is as follows: • Evaluate operations inside grouping symbols such as brackets. • Evaluate multiplication or division going from left to right. • Evaluate addition or subtraction going from left to right.

Mathspace WA Curriculum – Year 7 mathspace.co

1.04 Practice questions Understanding

Explorer

Adventurer

1–2

Trailblazer

1–2

What is the order of operations when performing arithmetic with whole numbers?

The ﬁrst step when calculating 2 × (7 + 3) is to do 7 + 3. What is the ﬁrst step in each of these problems? a

3×5+1

10 − 4 × 2

9÷3+5×2

7−2+6−1

Fluency Ex 1

3a-h, 4a-gi, 5, 8ac

Ex 3

6+4×2

8 × (8 − 6)

6 + 32 ÷ 4

24 ÷ 6 × 4

22 − 18 ÷ 6

37 − 35 ÷ 7 + 8

36 ÷ 4 × 8 + 6

18 ÷ 2 + 7

8×5−3×2

7+4×3−6

11 + 10 ÷ 5 × 8

Calculate: a

(7 + 9) × 10

7 × (5 + 4) − 50

34 − [29 + (12 ÷ 3)]

(8 − 2) × 8

(7 − 2) × (6 − 2)

(10 + 2) × (7 − 2)

[69 − (15 + 24)] ÷ 3 + 8 × 6

(11 + 5) ÷ (6 − 2)

[(8 + 2) × 6] ÷ 5

{[100 − (5 + 15)] + 8} ÷ 8

[7 × (6 − 3)] × [(8 + 2) ÷ 5]

[1 + (6 × 3 − 9)] ÷ [20 − (100 ÷ 10)]

m 8 × 0 ÷ 8 ÷ 1

12 × 2 ÷ 2 ÷ 3

Given the expression 6 + 19 − 4 × 3: a

Evaluate the expression.

These expressions are similar to the given expression, except that a pair of brackets has been added. Evaluate these expressions: i

3g-n, 4g-n, 7, 8bdfh

Calculate:

m Ex 2

3c-j, 4e-kn, 6, 8aceg

6 + (19 − 4) × 3

6 + 19 − (4 × 3)

iii

(6 + 19 − 4) × 3

Given the expression 6 × 15 − 5 + 3: a

Evaluate the expression.

These expressions are similar to the given expression, except that a pair of brackets has been added. Evaluate these expressions: i

6 × 15 − (5 + 3)

6 × (15 − 5) + 3

iii

(6 × 15) − 5 + 3

1.04 Order of operations with whole numbers mathspace.co

Given the expression 35 − 6 × 3 + 2: a

Evaluate the expression.

These expressions are similar to the given expression, except that a pair of brackets has been added. Evaluate these expressions: i

35 − (6 × 3) + 2

35 − (6 × 3 + 2)

iii

(35 − 6) × 3 + 2

35 − 6 × (3 + 2)

Insert a pair of brackets in the appropriate location to make these statements true. a

12 − 6 + 6 ÷ 6 = 10

5 + 10 × 4 − 5 × 4 = 40

4 + 5 × 13 − 5 − 4 = 40

5 × 15 − 4 ÷ 2 + 8 = 65

18 × 4 − 1 + 4 − 18 = 40

23 − 10 × 3 + 8 × 9 − 2 = 109

15 ÷ 3 + 7 × 4 + 2 × 6 = 257

30 − 4 × 5 ÷ 5 − 3 = 20

9, 11

10–12

Reasoning 9

Given the expression 8 + 3 × 6: a

Determine whether these expressions result in the same value as the given expression: i

b 10

3×6+8

8 + (3 × 6)

Determine whether these expressions result in the same value as the given expression: 7 ÷ (5 + 9)

9+7÷5

iii

5+9÷7

(7 ÷ 5) + 9

How do the brackets in 7 ÷ (5 + 9) change the order of operations from the given expression 7 ÷ 5 + 9?

Determine whether these expressions result in the same value as the given expression: i

58 − (7 × 8)

(58 − 7) × 8

iii

58 × 5 − 7

How do the brackets in (58 − 7) × 8 change the order of operations from the given expression 58 − 7 × 8?

Given the expression 7 ÷ 4 − 1: a

Determine whether these expressions result in the same value as the given expression: i

4−1÷7

(7 ÷ 4) – 1

iii

7 ÷ (4 − 1)

How do the brackets in 7 ÷ (4 − 1) change the order of operations from the given expression 7 ÷ 4 − 1?

Problem-solving 13ac, 14–15

13ace, 15–16

13bdf, 15–17

Fill in the boxes with the right operations to make these statements true. a c e

(8 + 3) × 6

Given the expression 58 − 7 × 8: a

iii

How do the brackets in (8 + 3) × 6 change the order of operations from the given expression 8 + 3 × 6?

6×8+3

Given the expression 7 ÷ 5 + 9: a

9–10

7 ⬚ 3 ⬚ 4 − 3 = 14

20 ⬚ 5 ⬚ 4 − 6 = 10

15 ⬚ 3 ⬚ 2 − 4 = 43

Mathspace WA Curriculum – Year 7 mathspace.co

b d f

12 ⬚ 3 ⬚ 2 × 4 = 288

9 × 2 ⬚ 5 ⬚ 3 − 1 = 32

30 × 6 ⬚ 3 ⬚ 2 = 58

A gardener has 5 bags with 12 apples in each bag. He discovers there are 2 rotten apples in each bag. How many good apples does he have?

Before going back to school a student purchases 4 pens costing $3 each and 5 pencils costing $2 each. Find the total spent by the student.

A teacher has 50 lollies in a bag and wants to give 3 lollies to each of her 16 students. How many lollies would the teacher have left?

Josie has 100 marbles in her collection. After picking out her favourite 12, she shared the remainder amongst her 4 friends. How many marbles does each friend receive?

1.05 Check reasonableness of calculations After this lesson, you will be able to… • estimate and check the reasonableness of an answer.

Check reasonableness To estimate, we can round numbers to make the calculation easier. The more we round, the less accurate the estimate will be. To ﬁnd a reasonable estimate, we can round to the nearest 10, 100 or other place value. 10

When rounding to the nearest 10, we are asking whether this number is closer to 10 or 20. In this case, we can see that 17 is closer to 20. So, 17 rounded to the nearest 10 is 20.

Example 1 Is this a reasonable statement? “67 − 29 is close to 40”?

Create a strategy

Apply the idea

Round each number to the nearest 10, then evaluate.

Rounding off each number to the nearest 10 would give us: 70 − 30, which is equal to 40. So, “67 − 29 is close to 40” is a reasonable statement.

1.05 Check reasonableness of calculations mathspace.co

Example 2 Is this statement accurate? “4 × 58 will be less than 200.”

Create a strategy Round the larger number up and down to the nearest 10 and evaluate the expression.

Apply the idea Rounding down 58 to the nearest 10 will give us 50. 4 × 50 = 200

Evaluate the multiplication

Rounding up 58 to the nearest 10 will give us 60. 4 × 60 = 240

Evaluate the multiplication

We know that 58 is larger than 50 and smaller than 60. So, we can say that 4 × 58 will be more than 200 but less than 240. So, the statement “4 × 58 will be less than 200” is not accurate.

Idea summary Round numbers to the nearest place value like 10, 100 or 1000. This will make the calculation easier and give a rough idea of what the answer should be.

1.05 Practice questions Understanding 1

Explorer 1aceg, 2aceg, 3a-dg, 4

Trailblazer 1bdfh, 2bdfh, 3d-h, 4

Round these numbers to the nearest 10: a

345

132

909

Round these numbers to the nearest 100: a

825

276

913

127

166

659

482

995

Round these numbers to the nearest 1000: a

1234

6890

9425

4765

9938

44 987

19 439

66 902

Consider the product of 37 × 94 and determine if each statement is correct or incorrect. Provide the correct statement if necessary. • The product is less than 4000. • The product is more than 2700. • The product is more than 4000. • The product is less than 2700.

Adventurer 1bdfh, 2bdgh, 3c-g, 4

Mathspace WA Curriculum – Year 7 mathspace.co

Fluency Ex 1

Ex 2

59 − 27 is approximately equal to 30.

68 − 27 is approximately equal to 60.

2883 − 2831 is approximately equal to 80 − 30.

6871 − 5832 is approximately equal to 70 − 30.

3 × 85 will be more than 240.

6+3+3

23 + 42 + 34

93 + 342 + 567

7+2+3

9 + 8 − 12

9+9−7

45 + 12 − 7

203 × 6

68 × 69

1621 ÷ 4

5513 ÷ 11

2 + 8 + 19

28 + 32 + 43

99 + 88 − 78

131 + 62 + 11

Which of these expressions are approximately equal to 1000? a

452 + 213 + 334

99 + 188 − 68

948 + 827 + 18

946 + 28 + 14 + 19

Determine whether these products are approximately equal to 1200: 33 × 37

31 × 18

Is 821 ÷ 400 approximately equal to 200?

Is 4 ÷ 1221 approximately equal to 300?

Approximate these calculations to the nearest 100: a

45 × 93

Reasoning

178 × 5

15, 17ab, 19cd, 21bd

15ac, 17bc, 19ad, 21ab

15ac, 16, 17cd, 18, 19ab, 20, 21ac, 22

If you estimate these sums, will the estimate be larger or smaller than the actual answer? a

364 + 292 + 311

Which of these expressions are approximately equal to 100?

Which of these expressions are approximately equal to 10?

a 10

4 × 89 will be less than 320.

Determine whether these sums are approximately equal to 10, 100 or 1000:

a 9

5b-d, 6b, 7cd, 9–11, 14

Are these statements accurate? Explain your choice.

a 8

5a-c, 6b, 7bc, 9, 12–14

Are these statements reasonable? Explain your choice.

a 7

5a-c, 6a, 7ab, 8, 12–14

63 + 75

314 + 78

198 + 47

987 + 332

When estimating sums, is it possible to make a generalisation about the size of the estimation compared to the actual answer when: a

both values are rounded up?

both values are rounded down?

one value is rounded up and the other down?

1.05 Check reasonableness of calculations mathspace.co

If you estimate these differences, will the estimate be larger or smaller than the actual answer? a

569 − 114

719 − 87

both values are rounded up?

both values are rounded down?

the larger value is rounded up and the smaller down?

the larger value is rounded down and the smaller up?

959 − 674

If you estimate these products, will the estimate be larger or smaller than the actual answer? 132 × 43

714 × 98

15 × 41

178 × 9

When estimating products, is it possible to make a generalisation about the size of the estimation compared to the actual answer when: a

both values are rounded up?

both values are rounded down?

one value is rounded up and the other down?

If you estimate these quotients, will the estimate be larger or smaller than the actual answer? 275 ÷ 29

148 ÷ 19

599 ÷ 63

125 ÷ 52

When estimating quotients, is it possible to make a generalisation about the size of the estimation compared to the actual answer when: a

both values are rounded up?

both values are rounded down?

the larger value is rounded up and the smaller down?

the larger value is rounded down and the smaller up?

Problem-solving 23, 26 23

a 22

When estimating differences, is it possible to make a generalisation about the size of the estimation compared to the actual answer when:

a 20

83 − 41

23, 25–26

24–26

Michael collects padlocks. He knows that he has exactly 195 of them. Michael has to move house this week and must buy storage boxes to transport them. Should Michael buy 7 boxes that hold 30 padlocks each or 5 boxes that hold 30 padlocks each?

Mathspace WA Curriculum – Year 7 mathspace.co

Ryan and Michael went out one evening. Michael said: “I have $17.40, which is about $20. This restaurant charges $9.75 for a pizza, which is about $10. That means I have enough money to pay for both of us to get a pizza.” a

Is Michael correct? Explain your answer.

Ryan says, “I can put $5 towards our meal, then we can both afford a pizza.” Is Ryan correct? Explain your answer.

A rectangular sticker is 59 mm by 112 mm. Explain how you could ﬁnd the approximate area of the sticker.

A school has 123 students in Year 7, 194 in Year 8, 211 in Year 9 and 49 in Year 10. Explain how you could ﬁnd the approximate number of students in the school.

Investigation: How long does it take to count to a million?

Investigate online

mathspace.co

1.06 Number calculations with technology After this lesson, you will be able to… • use technology to perform operations with whole numbers and check the reasonableness of your answer.

Operations with whole numbers using technology The most common type of technology used for performing operations is the hand-held calculator, so let’s run through the basics. 1.06 Number calculations with technology mathspace.co

The buttons we will use at the moment are: • The number buttons: 0 - 9

• The equals sign =

• The operations: + , - , × and ÷

• Round Brackets ( and )

To perform an operation between two numbers with your calculator: • Press the buttons corresponding to the digits of the ﬁrst number, going from left to right • Press the corresponding operation button • Enter the second number the same way as the ﬁrst • Press the equals sign The answer should now appear on the calculator display. If we are using a modern calculator, it will automatically follow the correct order of operations, but if not, we will have to do that for ourselves. It’s always useful to check the reasonableness of your answer. If the answer doesn’t seem right, there’s a good chance it isn’t. Here are some common calculator errors: • Missing a digit from a number • Swapping the digits • Using the wrong operation • Adding an extra digit

Example 1 Use a calculator to evaluate 214 + 443.

Create a strategy

Apply the idea

Press the calculator buttons in order of the digits and symbols in the expression, followed by the equals sign.

Press the buttons: 2 14 + 4 43 = to get: 214 + 443 = 657

Example 2 Using a calculator, evaluate 47 − (14 + (14 ÷ 7)).

Create a strategy

Apply the idea

Press the calculator buttons in order of the digits and symbols in the expression, followed by the equals sign.

We can press the calculator buttons according this sequence: 4 7 − (1 4 + (1 4 ÷ 7)) = to get: 47 − (14 + (14 ÷ 7)) = 31

Mathspace WA Curriculum – Year 7 mathspace.co

Idea summary The buttons we need to use on our calculators are: • The number buttons: 0 - 9 • The operations: + , − , × and ÷ • The equals sign = • Round Brackets ( and ) It’s always useful to check the reasonableness of your answer. If the answer doesn’t seem right, there’s a good chance it isn’t.

1.06 Practice questions Explorer

Understanding 1

507

429

507

429 ÷

Danielle wants to evaluate the expression (224 − 126) ÷ 14. In which order should Danielle push the buttons on her calculator to get the correct answer? A B

Trailblazer 1–4

Sandy wants to evaluate the expression (429 + 507) ÷ 13. In which order should Sandy push the buttons on her calculator to get the correct answer? A

1–4

Xanthe wants to evaluate the expression (89 + 74) × 33. In which order should Xanthe push the buttons on her calculator to get the correct answer? A

Adventurer

1–4

224

−

126

224

−

126

Rosey wants to evaluate the expression (789 − 31) × 24. In which order should Rosey push the buttons on her calculator to get the correct answer? A B

−

789 31

31 =

−

789 1.06 Number calculations with technology mathspace.co

Fluency 5 Ex 1

6 Ex 2

5c-f, 6b-d, 7c-g

5e-h, 6c-e, 7f-j

Evaluate: a

214 + 300

99 735 − 37 926

629 + 543

800 ÷ 10

2939 × 46

3563 ÷ 7

5135 ÷ 65

932 × 28

720 ÷ 7 × 7000

44 − (31 + (8 ÷ 2))

34 + 3294 + 909 + 6204

4 × 6 + 86 − 30

4 × 100 000 + 9 × 10 000 + 2 × 100 + 6 × 10

Evaluate: a

Determine whether these equations are true or false: a

7 × (3 + 3) = 14

8÷4×2=1

4 + (5 × 6) = 34

(9 + 7) × 10 = 79

(11 + 9) × 5 = 56

(10 × 7) × 2 = 140

17 − (2 + 4) = 19

23 − (15 − 9) = 17

28 − (16 ÷ 4) = 3

(100 ÷ 10) ÷ 2 = 20

Reasoning 8

5a-d, 6a-c, 7a-e

8–10

9–12

11–14

Xavier used his calculator to evaluate 18 × 75 − 8 and got the answer 1018. Is Xavier’s answer correct? If not, explain what he might have done wrong.

Lucy used her calculator to evaluate (20 − 18) × 3 + 19 and got the answer 25. Is Lucy’s answer correct? If not, explain what she might have done wrong.

Maximilian used his calculator to evaluate 3 × 700 and got the answer 210. Is Maximilian’s answer correct? If not, explain what he might have done wrong.

Vincent used his calculator to evaluate 6 × 40 000 and got the answer 2 400 000. Is Vincent’s answer correct? If not, explain what he might have done wrong.

Michael used his calculator to evaluate (2 + 50) ÷ 77 and got the answer

Is Michael’s answer correct? If not, explain what he might have done wrong. 13

Lucy used her calculator to evaluate 20 − (5 + 14) ÷ 21 and got the answer

Is Lucy’s answer correct? If not, explain what she might have done wrong. 14

Brad used his calculator to evaluate (4 × 11 + 9) ÷ 14 − 20 and got the answer

Is Brad’s answer correct? If not, explain what he might have done wrong.

Problem-solving 15–17 15

15–17

A charity group for tsunami relief has gathered 588 first aid kits. 21 shelters have been affected. The charity decided to distribute the first aid kits equally between these. How many first aid kits does each shelter receive?

18–19

Mathspace WA Curriculum – Year 7 mathspace.co

A cargo ship is delayed by a storm at sea for two weeks. When it arrives, the crew finds that 3 of the 81 crates of fruit have spoiled due to the delay. They decide to share the remaining good crates equally among their 13 customers, with any extras sold separately. a

Find the number of fruit crates that each customer will receive.

Each crate contained 32 kg of fruit and each kilogram of fruit could be sold for $4. Calculate the loss incurred by the crew due to the spoilage.

A teacher has ﬁfteen bags of lollies. Each bag contains 18 lollies but the teacher eats three lime lollies from each bag before sharing the remainder amongst her 25 students. How many lollies will each student receive?

Gus wants to give away his 120 marbles to his friends. He is going to give half the marbles to Corey. Corey will get three times as many as Alﬁe and Barry will get twice as many as Alﬁe. How many marbles does each friend receive?

In a class of 10 students, two have 2 siblings, four have 1 sibling and one is an only child. If there are 17 siblings in total, how many students have 3 siblings?

1.07 Write mathematical statements After this lesson, you will be able to… • translate written text into mathematical expressions using the four operations of addition, subtraction, multiplication and division. • describe the relationship between numbers using symbols such as greater than, less than, greater than or equal to, less than or equal to, equal to and not equal to.

Four operations Let’s start with the four operations. These symbols tell us what to do with our numbers and are usually referred to as: plus, minus, times and divide. Word description Sum of Difference between Product of Quotient of

Mathematical operation Addition (plus) Subtraction (minus) Multiplication (times) Division (divide)

Mathematical symbol + − × ÷

1.07 Write mathematical statements mathspace.co

There are also other ways we can refer to them. decrease

increase sum

deduct

take away

total

plus

less than

minus

subtract

add

difference

more than

Operations

groups of

quotient

multiply

divide

ratio

times

split

product

Here are a few ways that we can refer to the same operation using different words: • 4 + 9 can be described by: “the sum of 4 and 9” or “4 plus 9” or “9 more than 4”. • 8 – 3 can be described by: “the difference between 8 and 3” or “8 minus 3” or “3 less than 8”. • 5 × 6 can be described by: “the product of 5 and 6” or “5 times 6” or “6 groups of 5”. • 9 ÷ 3 can be described by: “the quotient of 9 and 3” or “9 divided by 3” or 9 shared between 3”. Notice that when writing “8 − 3” as “3 less than 8” the order of the numbers switches around. This is important to remember because “8 − 3” and “3 − 8” have different solutions.

Example 1 Which expression describes ‘five groups of six’? A 6−5

6×5

6+5

Create a strategy

Apply the idea

The phrase “groups of” indicates multiplication.

5 × 6 is the same as 6 × 5.

6÷5

So the answer is option B: 6 × 5

Mathspace WA Curriculum – Year 7 mathspace.co

Idea summary A summary of terms that can be used in mathematical statements to describe the four operations: decrease

increase sum

deduct

take away

total

plus

less than

minus

subtract

add

difference

more than

Operations

groups of

quotient

multiply

divide

ratio

times

product

split

Equality and inequality symbols In addition to the four operations, we also have symbols to describe the relationship between numbers. When comparing two values or expressions, if they are equal in value, they are considered to be in a state of equality. The symbol “ = ” is used to denote equality. In contrast, inequality describes the state where two values or expressions are not the same. This can occur in various forms, such as one value being greater than, less than, greater than or equal to or less than or equal to another value. Word description

Symbol

Example

Greater than

“5 is greater than 2” can be written as “5 > 2”

Less than

“3 is less than 7” can be written as “3 < 7”

Greater than or equal to

≥

“5 is greater than or equal to 4” can be written as “5 ≥ 4”

Less than or equal to

≤

“6 is less than or equal to 7” can be written as “6 ≤ 7”

Equal to

“4 is equal to 4” can be written as “4 = 4”

Not equal to

≠

“4 is not equal to 5” can be written as “4 ≠ 5”

Let’s look at how we can use these symbols to represent a word statement in the next example.

1.07 Write mathematical statements mathspace.co

Example 2 Write this word statement using mathematical symbols: “nineteen is greater than eleven plus six”.

Create a strategy

Apply the idea

Replace each word or phrase with its corresponding mathematical symbol.

19 > 11 + 6

Idea summary We have some symbols to describe the relationship between numbers: Word description

Symbol

Greater than

Less than

Greater than or equal to

≥

Less than or equal to

≤

Equal to

Not equal to

≠

1.07 Practice questions Understanding 1

Trailblazer 2

Addition

Subtraction

Multiplication

Match each word description with the correct mathematical symbol: a

Adventurer 2

List three other words that have the same meaning as: a

Explorer 1–2

Greater than

−

Less than or equal to

Equal to

iii

Sum

≠

Less than

Difference

≤

Greater than or equal to

vii ×

Product

viii ≥

Not equal to

Quotient

Mathspace WA Curriculum – Year 7 mathspace.co

Division

Fluency Ex 1

Ex 2

The sum of 2 and 7

The difference between 7 and 6

The product of 3 and 10

The quotient of 12 and 4

Four groups of seven

Thirteen is not equal to eleven

4 is less than 10

8 is greater than 5

Write these word statements using mathematical symbols: Fifteen is equal to seven plus eight

Six is equal to sixteen minus ten

Eighty is equal to eight times ten

Five is equal to ﬁfty divided by ten

Nineteen is greater than seven plus ﬁve

Write these word statements using mathematical symbols: a

The difference of 24 and 8, divided by 4

12 times the result of 10 reduced by 6

1 is equal to 12 divided by 3 minus 3

11 times 2 take away 9 is not equal to 10

15 is less than the quotient of 100 and 5

20 is greater than or equal to 2 groups of the sum of 3 and 6

Which option matches the statement “8 more than the quotient of 9 and 3”? B

(2 × 3) − (10 + 12)

(10 − 12) + (2 × 3)

(10 + 12) − (2 × 3)

14 = 7 × (10 − 8)

14 = 7 × 10 − 8

14 = 7 + (10 − 8)

14 = 7 − (10 − 8)

5 × 4 < 25

5 × 4 > 25

5 × 4 ≤ 25

5 × 4 ≥ 25

Which option matches the statement “10 is greater than or equal to double the difference between 11 and 7”? A

(2 × 3) + (10 + 12)

Which option matches the statement “The product of 5 and 4 is less than 25”? A

Which option matches the statement “14 is equal to 7 groups of the difference between 10 and 8”? A

Which option matches the statement “1 is 5 less than the quotient of 24 and 4”? A

Which option matches the statement “The product of 2 and 3 is subtracted from the sum of 10 and 12”? A

3e-h, 4e, 5aef, 8–12

A 7

3c-f, 4e, 5aef, 6–7, 10–11

Write these word statements using mathematical symbols:

3a-d, 4–8

10 ≤ 2 × (11 − 7)

10 ≥ 2 × 11 − 7

10 ≤ 2 × 11 − 7

10 ≥ 2 × (11 − 7)

Given the statement: “The sum of 2 and the quotient of 80 and 10”: a

Write this statement using mathematical symbols.

Find the ﬁnal value of the statement.

1.07 Write mathematical statements mathspace.co

Reasoning 13

13ad

13ab

13bc

Some students wrote 4 stories to explain the expressions. Unfortunately, they are all incorrect. Identify the error and rewrite the story. a

Expression: 15 − 2 × 4

Incorrect story: There were 15 apples taken away from a box that 2 groups of 4 apples were in. b

Expression: 10 ÷ 2 + 3

Incorrect story: There were 10 cookies and 2 cookies were shared among 3 people. c

Expression: (10 − 4) ÷ 2

Incorrect story: There were 10 cupcakes that were divided between 2 people. One of them ate 4 of the cakes. d

Expression: 8+3×2

Incorrect story: There were 8 pencils and 3 more pencils were added, this group was then doubled.

Problem-solving 14

14–15

14–17

A car rental company charges $30 per day plus an additional $0.25 per kilometre driven. If a customer rents a car for 5 days, write a mathematical statement that would help you ﬁnd the maximum number of kilometres they can drive without exceeding a total rental cost of $425.

Suppose the average score of a class of 20 students in a test is 75. If a new student joins the class and the class average increases to 76, write a mathematical statement that would help you ﬁnd what score the new student achieved.

Alex and Beth play on the same hockey team. Today Alex scored 4 more goals than Beth but together they scored 12 goals. Write a mathematical statement that would let you ﬁnd the number of goals Alex scored.

A square has a perimeter of 20 m. Write a mathematical statement that would show how you could calculate the area of the square.

Mathspace WA Curriculum – Year 7 mathspace.co

MODELLING INVESTIGATION

Charity fundraising Explore how mathematics can help you plan a school fun run to raise money for a charity. Initially, you will be asked to decide on the track length you will use, then estimate how long it takes to jog a lap of your track and finally calculate how many laps can be completed in 1 hour. You will use this information to determine how much you would raise.

Objectives • To analyse the fun run situation and decide what is needed to plan it. • To use mathematics to show how many laps you can run and how much money you will raise. • To explain your plan and how your choices affect the outcome.

Analyse the fun run You are planning a school fun run that will last for 1 hour. To begin, you will need to determine some key attributes. You will need to determine the track length, estimated time to complete each lap and a donation amount per lap. Using these factors we will estimate how much money might be able to be raised.

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Investigate 1.

My track length is metres (for example 200 m up to 500 m). My estimate of how long to jog 1 lap is (for example 1 to 3 minutes). My estimate of how much a student may receive as a donation for each lap is . Explain how you determined each value.

Will you use exact numbers or approximate ones? Why?

What might affect your plan? List one assumption and one thing that limits you.

Represent with mathematics Using your determined values you will now calculate an estimate of how much money you can raise. Come up with a model that uses the attributes, track length, time per lap and donation per lap. Then use your determined values to calculate your charity fundraising amount.

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Investigate 1.

What is your generalised model?

What fundraised amount do you calculate using your model and the values you chose?

Consider the assumptions and limitations of this model and how it affects your fundraised amount.

Interpret your plan Explore what happens with your model when the estimated values are adjusted and what might happen if assumptions are changed.

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Adjust one assumption or estimate value (for example add 10 minutes practice time or use 20 cents less per lap), then recalculate and compare the results.

Investigate 1.

Do your number of laps and money raised seem realistic for 1 hour? Why or why not?

Change one assumption of estimated value. How does it change your total?

Explain your fun run plan to the class. How do your assumptions affect it?

Discussion

What was the hardest part of planning the fun run with mathematics? Why?

How could you make your plan better if you had more time or helpers?

Talk to a friend about their plan. How was it different from yours?

Mathspace WA Curriculum – Year 7 mathspace.co

1 Chapter review 1

Insert a pair of brackets in the appropriate location to make these statements true: a

7 × 3 + 9 − 12 + 2 = 20

5 + 10 × 4 − 5 × 4 = 40

9 + 6 ÷ 3 × 4 − 8 = 12

14 ÷ 2 × 4 + 9 − 3 × 2 = 31

Given the expression 4 + 11 − 3 × 2: a

Evaluate the expression.

These expressions are similar to the given expression, except that a pair of brackets has been added. Evaluate these expressions: i

4 + (11 − 3) × 2

4 + 11 − (3 × 2)

iii

(4 + 11 − 3) × 2

Fill in the boxes with the right operation to make these statements true: a c

7 ⬚ 3 ⬚ 6 − 3 = 22

20 ⬚ 2 ⬚ 5 + 3 = 11

8 ⬚ 3 ⬚ 6 − 2 = 28

9 × 2 ⬚ 5 ⬚ 3 − 1 = 32

A student purchases 5 boxes of crayons costing $7 each and 8 sketch pads costing $5 each. Find the total spent by the student.

Round these numbers to the nearest 1000:

1264

8890

7425

2234

9838

43 976

18 439

69 902

Determine whether these statements are reasonable and explain your choice. a

84 + 29 is approximately equal to 110.

76 + 15 is approximately equal to 100.

9134 + 2678 is approximately equal to 9000 + 3000.

7652 − 1389 is approximately equal to 7500 − 1000.

Consider the product of 47 × 93 and determine which of these statements are correct: a

The product is less than 5000.

The product is more than 3600.

The product is more than 5000.

The product is less than 3600.

Determine whether these sums are approximately equal to 10, 100 or 1000: a

29 + 46 + 32

83 + 442 + 667

394 + 292 + 321

5513 ÷ 11

Which of these expressions are approximately equal to 1000? a

652 + 243 + 334

45 + 188 − 68

648 + 227 + 18

646 + 281 + 14 + 19

Approximate these calculations to the nearest 100: a

6+2+3

45 × 93

178 × 5

1621 ÷ 4

Hiro wants to evaluate the expression (24 + 83) × 56. In which order should Hiro push the buttons on his calculator to get the correct answer? A

+ × 24 83 =

Chapter 1 review mathspace.co

Determine whether these equations are true or false: a

3 + (4 × 5) = 34

(9 + 7) × 10 = 160

(11 + 9) × 5 = 100

23 − (15 − 9) = 23

28 − (16 ÷ 4) = 24

(100 ÷ 10) ÷ 2 = 5 g

15 × (3 + 3) = 14

8 ÷ 4 × 2 = 15

As a result of a tsunami, 22 shelters have been setup to support affected communities. A charity group for tsunami relief has gathered 638 blankets and decides to distribute the blankets equally between the shelters. How many blankets does each shelter receive?

A cargo ship encounters a storm at sea and is delayed by two weeks. Once it reaches its destination, the crew discover that 6 out of the 66 crates of fruit that it carried have spoilt. The unspoilt crates will be divided equally among their 20 customers. a

Find the number of crates of fruit each customer will receive.

Each crate contained 32 kg of fruit and each kilogram of fruit could be sold for $4. Calculate the loss incurred by the crew because of the spoilage.

A teacher has twelve bags each containing 22 lollies. The teacher eats three lime lollies from each bag before sharing the remainder amongst her 19 students. How many lollies will each student receive?

Write these word statements using mathematical symbols: a

The sum of 2 and 7

The difference between 7 and 6

The product of 3 and 10

The quotient of 12 and 4

Four groups of seven

Thirteen is not equal to eleven

4 is less than 10

8 is greater than 5

2 × 7 = 14

Write these equations using word statements: a

12 + 8 = 20

10 ÷ 5 ≠ 5 ÷ 10

8−1=7

Mathspace WA Curriculum – Year 7 mathspace.co

36 ÷ 4 = 9

Write these word statements using mathematical symbols: a

The difference of 24 and 8, divided by 4

12 times the result of 10 reduced by 6

1 is equal to 12 divided by 3 minus 3

11 times 2 take away 9 is not equal to 10

15 is less than the quotient of 100 and 5

20 is greater than or equal to 2 groups of the sum of 3 and 6

Given the statement: “The sum of 2 and the quotient of 80 and 10”: a

Write this statement using mathematical symbols.

Find the final value of the statement.

Alex and Beth play on the same hockey team. Today, Alex scored 4 more goals than Beth but together they scored 12 goals. Write a mathematical statement that would let you find the number of goals Alex scored.

A square has a perimeter of 20 m. Write a mathematical statement that would show how you could calculate the area of the square.

A taxi charges $25 per day plus an additional $0.25 per kilometre driven. If a customer rents a car for 8 days, write a mathematical statement that would help you find the maximum number of kilometres they can drive without exceeding a total rental cost of $325.

Did you know?

Thanks to technology, large whole-number calculations that would take humans hours, days or even years to compute can be done in the blink of an eye. Calculators and computers can perform millions of operations per second, allowing mathematicians, engineers and scientists to solve complex problems efficiently. It’s amazing how far we’ve come from using just pen and paper!

Chapter 1 review mathspace.co

Big ideas • Basic rules govern number manipulation, including associative, commutative and distributive laws, which support accurate mathematical problem-solving. • Mastery of the concepts of factors, multiples and prime factorisation enables determination of the highest common factor and least common multiple of numbers. Utilisation of these concepts in solving problems that involve shared properties and relationships between numbers.

Curriculum Outcomes • Represent in expanded form, evaluate and compare numbers expressed in index notation, including powers of 10 • Extend knowledge of factors to represent natural numbers as products of prime factors using index notation as appropriate • Explore and explain connections between square numbers and square roots, cube numbers and cube roots, as products of repeated factors • Extend the use of associative, commutative and distributive laws, additive and multiplicative partitioning, inverse operations, order of operations, equality and inequality to validate a range of mental and written strategies involving the four operations on whole numbers, positive fractions and decimals, and addition and subtraction of integers

2 Powers, factors and multiples Chapter outline 2.01 Number operation properties 2.02 Square and cube numbers 2.03 Square and cube roots 2.04 Powers of whole numbers 2.05 Powers of ten 2.06 Factors and multiples Investigation: The sieve of Eratosthenes 2.07 Prime factorisations Investigation: Celtic knots 2.08 HCF and LCM Chapter 2 review

46 55 62 68 73 78 83 89 97

A million is pretty huge – it has six zeros! How long do you think it takes a million seconds to pass?

2.01 Number operation properties After this lesson, you will be able to… • apply the associative law, commutative law and distributive law to perform mental calculations and rearrange numbers to make calculations easier.

The associative law The associative law tells us that when we add or multiply any three numbers, the grouping (or association) of the numbers does not affect the answer. For example, 1 + 2 + 3 is the same as (1 + 2) + 3 and 1 + (2 + 3). So, we can add pairs of numbers together in whichever order is easiest. Another example of the associative law for addition is shown: (4 + 7) + 3

4 + (7 + 3)

The same works for multiplication, since 2 × 3 × 5 is the same as (2 × 3) × 5 and 2 × (3 × 5). The associative law doesn’t apply to subtraction or division, where we should always work from left to right. For example: (22 − 7) − 5 ≠ 22 − (7 − 5) and (24 ÷ 6) ÷ 2 ≠ 24 ÷ (6 ÷ 2).

Example 1 Which of these expressions is equal to 19 + (38 + 25)? A 19 + 28 + 25

(19 + 38) + 25

C (19 + 38) − 25

(19 × 38) + (19 × 25)

Create a strategy Identify the option that utilises the associative law where we can group the numbers in different ways as long as we keep the order of addition the same.

Mathspace WA Curriculum – Year 7 mathspace.co

Apply the idea Options C and D have different operations: subtraction and multiplication, respectively. Option A involves adding 28 and not 38. This means that the answer is option B: (19 + 38) + 25

Idea summary The associative law states that we can group addition or multiplication in any order and the answer stays the same. For example: (4 + 5) + 6 = 4 + (6 + 5).

The commutative law The commutative law tells us that we can swap the order of numbers around and still get the same answer when we add or multiply them. For example, 4 + 3 = 3 + 4, so the order of addition doesn’t affect the ﬁnal sum. The same is true for multiplication. 4 × 3 = 3 × 4; the order doesn’t affect the product. This really helps when we have to multiply several numbers together. Another example of the commutative law for multiplication is shown: 7×4

4×7

The commutative law only applies to adding and multiplying, not subtracting and dividing, where the order is important. For example: 10 − 5 ≠ 5 − 10 and 8 ÷ 4 ≠ 4 ÷ 8.

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Sometimes reordering the terms in a question can make life easier. Which seems to be the easiest way to work out 2 × 13 × 5? 2 × 13 × 5

2 × 5 × 13

26 × 5

10 × 13

130

Both give the same answer, but by reordering the terms, the second may seem easier because we are now multiplying by 10. The same works for addition: 13 + 18 + 7

13 + 7 + 18

31 + 7

20 + 18

Again, both give the same answer, but the second may seem easier because we bridged to the nearest 10 first.

Example 2 Given the expression 4 × 13 × 5: a Which pair of numbers will be easiest to multiply together ﬁrst? A

4×5

13 × 5

4 × 13

Create a strategy

Apply the idea

Choose the smaller pair of numbers that have a product that is a multiple of 10.

The smallest pair of numbers are 4 and 5, and their product is a multiple of 10. Answer is option A: 4 × 5.

b Complete the working out. 4 × 13 × 5 = 4 × ⬚ × ⬚ =⬚×⬚

Create a strategy

Apply the commutative law Evaluate the ﬁrst multiplication

=⬚

To apply the commutative law, rearrange the expression and write the pair of numbers that are easiest to multiply ﬁrst. In this case, we want to use the pair found in part (a).

Apply the idea 4 × 13 × 5 = 4 × 5 × 13

Apply the commutative law

= 20 × 13

Evaluate the ﬁrst multiplication

= 260

Evaluate

Mathspace WA Curriculum – Year 7 mathspace.co

Example 3 Given the expression 27 + 48 + 13: a Which pair of numbers will be easiest to add together ﬁrst? A

48 + 13

27 + 13

27 + 48

Create a strategy Choose the pair of numbers where the ones will sum up to 10 or less.

Apply the idea Option A 8 + 3 = 11 Option B 7 + 3 = 10 Option C 7 + 8 = 15 Answer is option B: 27 + 13.

b Find the missing value of the expression to complete the working out. 27 + 48 + 13 = 27 + ⬚ + ⬚ =⬚+⬚

Create a strategy

Apply the reordering Evaluate the ﬁrst addition

=⬚

Use the associative law.

Apply the idea 27 + 48 + 13 = 27 + 13 + 48

Apply the associative law

= 40 + 48

Evaluate the ﬁrst addition

= 88

Evaluate

Idea summary The commutative law states that we can write addition or multiplication in any order and the answer will stay the same. For example: 7 × 8 = 8 × 7. In reordering, questions can be done in a different order to make the questions easier. For example: 13 + 18 + 7 = 13 + 7 + 18

2.01 Number operation properties mathspace.co

The distributive law This is a method we will explore more later when we learn algebra, but it can also help us to perform trickier multiplication questions. For example, if we want to multiply 9 by 43, we could rewrite the larger number 43 as the sum (40 + 3). The multiplication then becomes 9 × (40 + 3). We can then distribute the multiplication over each number in the brackets, like this: 9 × (40 + 3) = (9 × 40) + (9 × 3)

Using the distributive law, we can now expand this as (9 × 40) + (9 × 3), both of which are a lot easier to calculate. We can use a similar strategy to make division questions easier, like 168 ÷ 14. We know that 168 is 140 + 28, which we know are multiples of 14. So we can do: 168 ÷ 14 = (140 + 28) ÷ 14 = (140 ÷ 14) + (28 ÷ 14) = 10 + 2 = 12

Example 4 Given the expression 168 ÷ 7: a Complete the statement: 168 ÷ 7 is the same as (140 + ⬚) ÷ 7

Create a strategy

Find the number to be added to 140 that will give a sum of 168.

Apply the idea 140 + 28 = 168 168 ÷ 7 is the same as (140 + 28) ÷ 7. b Complete the statement: (140 + 28) ÷ 7 is the same as 140 ÷ 7 + ⬚ ÷ 7

Create a strategy

Consider the ﬁrst expression. Find the other number that is being divided by 7.

Apply the idea In (140 + 28) ÷ 7, both 140 and 28 will be divided by 7. (140 + 28) ÷ 7 is the same as 140 ÷ 7 + 28 ÷ 7.

Mathspace WA Curriculum – Year 7 mathspace.co

c Which arithmetic rule explains the equality between 168 ÷ 7 and 140 ÷ 7 + 28 ÷ 7?

Apply the idea From the previous parts we have seen: 168 ÷ 7 = (140 + 28) ÷ 7

168 was separated into two smaller numbers

= 140 ÷ 7 + 28 ÷ 7

The round brackets were distributed

Distributive law is the arithmetic rule that explains the equality. d Find the missing value of the expression to complete the working out. 168 ÷ 7 = (140 + 28) ÷ 7

Apply the idea

Separate into two manageable components

= 140 ÷ 7 + 28 ÷ 7

Distribute the round brackets using the distributive law

=⬚+⬚

Evaluate the division

=⬚

Evaluate

168 ÷ 7 = (140 + 28) ÷ 7

Separate into two manageable components

= 140 ÷ 7 + 28 ÷ 7

Distribute the round brackets using the distributive law

= 20 + 4

Evaluate the division

= 24

Evaluate

Idea summary The distributive law states that we can break apart larger numbers to help us calculate larger multiplications or divisions. For example: 1326 ÷ 13 = (1300 + 26) ÷ 13 = (1300 ÷ 13) + (26 ÷ 13).

2.01 Practice questions Understanding 1

Explorer 1, 3–4

Adventurer 1, 3–4

Trailblazer 1–4

Match these terms to their corresponding deﬁnitions: i

Associative law

It states that the grouping of numbers in an addition or multiplication expression does not change its value.

It rearranges the order of the numbers in an expression and the answer stays the same.

It states that changing the order of the addends does not change the sum or changing the order of the factors does not change the product.

Commutative law

iii

Reordering

2.01 Number operation properties mathspace.co

How does the distributive law help simplify calculations? Provide an example.

Is each expression an example of the associative law, commutative law, reordering or distributive law? a

15 + 5 = 5 + 15

54 ÷ 9 ÷ 3 = 54 ÷ 3 ÷ 9

24 + 6 + 2 = 24 + (6 + 2)

126 ÷ 3 = (120 + 6) ÷ 3

(15 + 6) × 4 = (15 × 4) + (6 × 4)

52 − 12 − 24 = 52 − 24 – 12

48 ÷ 2 ÷ 8 = 48 ÷ 8 ÷ 2

Are these statements true or false? a

4 × (3 + 5) = 4 × 3 + 4 × 5

4 + (3 × 5) = 4 × 3 + 4 × 5

(4 × 3) + 5 = 4 × 5 + 3 × 5

(4 + 3) × 5 = 4 × 5 + 3 × 5

5–12, 14, 16–17, 20–21

5–7, 10, 13, 15, 18–19, 21–22

(67 − 19) × 37

67 − 37 − 19

67 − (37 − 19)

35 + 29 + 11 = 35 + (⬚ + ⬚)

Given the expression 2 × 16 × 5: a

Which pair of numbers will be easiest to multiply together ﬁrst?

Complete the statement and then evaluate it: 2 × 16 × 5 = ⬚ × ⬚ × 16

Evaluate 17 + 39 + 23 using reordering.

67 × 37 − 67 × 19

28 + 46 − 16 = 28 + (⬚ − ⬚)

Fill in the boxes to complete the working out using the associative law. 13 × 4 × 5 = ⬚ × (4 × ⬚)

Fill in the boxes to complete the working out using the associative law. a

Ex 2

5–12, 14, 16–17, 20

Which of these expressions is equal to 67 − 19 − 37? A

24 × 6 = 6 × 24

Fluency Ex 1

Helen rearranged the numbers using reordering to make the addition easier. Complete the statement and then evaluate it: 24 + 38 + 16 = ⬚ + 16 + ⬚ Derek is evaluating the subtraction as shown. He noticed that it would be easier to subtract 27 from 47 ﬁrst and rearrange the numbers using reordering. Complete the statement and then evaluate it: 47 − 18 − 27 = ⬚ − 27 − ⬚ Given the expression 56 − 18 − 26: a

Which of the two numbers will be easier to subtract from 56 ﬁrst?

Complete the statement and then evaluate it: 56 − 18 − 26 = 56 − ⬚ − ⬚

Evaluate 57 − 29 − 17 using reordering.

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 3

Given the expression 54 + 26 + 38: a

Which pair of numbers will be easiest to add together ﬁrst?

Fill in the boxes to complete the working out. 54 + 26 + 38 = (⬚ + ⬚) + 38

Evaluate 4 × 17 × 5 using regrouping.

Given the expression 35 × 23: a

c 17

Ex 4

Evaluate the expression.

Complete the statement:

26 × 29 is the same as 26 × (⬚ − 1). 26 × (30 − 1) is the same as 26 × 30 − 26 × ⬚.

Evaluate the expression.

9 × 998

7 × 1002

11 × 124

6 × 412

Fill in the boxes to complete his working out, then evaluate the expression. 132 ÷ 11 ÷ 4 = 132 ÷ ⬚ ÷ 11

Given the expression 192 ÷ 6:

Fill in the box to complete the sentence.

192 ÷ 6 = (180 + ⬚) ÷ 6. (180 + 12) ÷ 6 is the same as 180 ÷ 6 + ⬚ ÷ 6.

Evaluate the expression.

Given the expression 119 ÷ 7: a

c 22

35 × 23 is the same as 35 × (20 + ⬚). 35 × (20 + 3) is the same as 35 × ⬚ + 35 × 3.

Dave is evaluating 132 ÷ 11 ÷ 4. He noticed that it would be easier to divide 132 by 4 ﬁrst and rearrange the numbers using reordering.

Complete the statement:

Evaluate: a

Given the expression 26 × 29: a

Complete the statement:

Fill in the box to complete the sentence.

119 ÷ 7 is the same as (140 − ⬚) ÷ 7. (140 − 21) ÷ 7 is the same as 140 ÷ ⬚ − 21 ÷ ⬚.

Evaluate the expression.

Evaluate using the distributive law: a

1616 ÷ 8

396 ÷ 4

522 ÷ 6

585 ÷ 13

2.01 Number operation properties mathspace.co

Reasoning

24–25

23–24

Using an example, explain why the commutative law does not work for division.

Julietta solved the problem as shown: 1

6 × 2 − 1 + 9 = 12 − 1 + 9

= 12 − (1 + 9)

Associative law

= 12 – 10

Evaluate the addition

Evaluate the subtraction

Evaluate the multiplication

Identify and explain Julietta’s mistake. 25

Describe the most eﬃcient method to evaluate 5 × 34 × 200.

Problem-solving 26 26

26–27

27–28

The area of a rectangle is equal to the length multiplied by width. Consider this rectangle: a

Find the area of Area 1.

Find the area of Area 2.

Find the total area.

Write the area calculation of 8 × 22 using the distributive law.

Area 1

Area 2

Consider this rectangle: a

Find the total area.

Find the non-shaded area.

Find the shaded area.

Write the area calculation of 16 × 20 using the distributive law.

Stefano wants to buy 4 shirts that cost $15 each and 3 pairs of pants that cost $25 each. If the store is offering a 10% discount on all purchases over $100, how much will the discount be? Use the distributive law to simplify the calculation.

Mathspace WA Curriculum – Year 7 mathspace.co

2.02 Square and cube numbers After this lesson, you will be able to… • write powers using the base and the index. • identify and calculate perfect squares and perfect cubes.

Square numbers A square number is a number that can be obtained by multiplying any number by itself. For example, 4 is a square number because it can be obtained by multiplying 2 by itself: 2×2=4 To indicate that a number has been multiplied by itself, we write the number being multiplied, called the base and then indicate the number of times it is multiplied by itself by writing this number, called the index (also known as the exponent or power), above and to the right of the base, like so: baseindex For example, if we have 3 × 3, we can see that 3 is being multipled by itself 2 times, so 3 is the base and the index is 2: 3 × 3 = 32 This can be read as “three to the power of two” or “three raised to the power of 2.” For now, we will just be looking at numbers raised to the power of two, resulting in perfect squares. In the above example 32 can also be referred to as “three squared” and results in the perfect square of 9. Let’s look at the pattern to see what it means to ‘square’ a number: 12

You can see that each formation of dots forms a perfect square. 12 = 1 × 1 = 1 22 = 2 × 2 = 4 32 = 3 × 3 = 9 42 = 4 × 4 = 16 and so on. 2.02 Square and cube numbers mathspace.co

Interactive exploration

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Example 1 Evaluate 92

Create a strategy Use the fact that squaring a number means multiplying by itself.

Apply the idea 92 = 9 × 9 = 81

Multiply the base by itself Evaluate

Example 2 Evaluate 42 + 92

Create a strategy Use the fact that squaring a number means multiplying by itself.

Apply the idea 42 + 92 = 4 × 4 + 9 × 9

Multiply the base by itself

= 16 + 81

Evaluate the multiplication

= 97

Evaluate

Example 3 Evaluate

Create a strategy Identify the order of operations.

Mathspace WA Curriculum – Year 7 mathspace.co

mathspace.co

Apply the idea Evaluate the square

Evaluate the subtraction

Evaluate the division

Evaluate the addition

Idea summary Perfect squares can be obtained by raising a smaller number to the power of two or multiplying a number by itself. When writing powers, the number being multiplied is called the base, the number of times it is multiplied by itself is called the index. In the example below 3 is the base and 2 is the index. 3 × 3 = 32

Cube numbers A cube number is a number that can be obtained by multiplying the same number by itself two more times. For example, 27 is a cube number because it can be obtained by multiplying 3 by itself two more times: 3 × 3 × 3 = 27 Numbers like 8, 27 and 64 are often called perfect cubes. They all come from multiplying the same number by itself two more times (raising it to the power of 3). In the table, we can see the first few cube numbers: Cube numbers 1 8 27 64 125 216 343 512 729 1000

Expanded form 1×1×1 2×2×2 3×3×3 4×4×4 5×5×5 6×6×6 7×7×7 8×8×8 9×9×9 10 × 10 × 10

Index form 13 23 33 43 53 63 73 83 93 103

2.02 Square and cube numbers mathspace.co

Example 4 Evaluate 53 + 32

Create a strategy Expand each by multiplying 5 by itself two more times while 3 by itself once.

Apply the idea 53 + 32 = 5 × 5 × 5 + 3 × 3

Expand each number

= 125 + 9

Evaluate the multiplication

= 134

Evaluate

Idea summary Perfect cubes can be obtained by multiplying a number by itself two more times.

2.02 Practice questions Understanding 1

72 A

7×7

2×2×2×2×2×2×2

7×2

4×3

4 × 4 × 4

3×3×3×3

What is the index?

What is the base?

Determine whether each number is perfect square, perfect cube, neither or both: a

121

324

512

729

Explain what a square number and a cube number are.

How can you determine whether a number is a square number?

Trailblazer 1–2, 3e-h, 4–5

Consider the expression 92. a

Adventurer 1–2, 3c-f, 4

Identify which one is equal to each expression: a

Explorer 1–2, 3a-d

Mathspace WA Curriculum – Year 7 mathspace.co

Fluency Ex 1

Ex 3

Ex 4

6cd, 7cd, 8bd-h, 9e-h, 10e-h, 11e-h, 12–13

122

232

2002

133

403

Evaluate: a

Ex 2

6bc, 7b, 8b-eg, 9c-f, 10cd, 11d-f, 12

Evaluate: a

6ab, 7a, 8a-eg, 9a-d, 10ab, 11a-d, 12

Evaluate: a

32 + 72

1002 + 102

92 − 82

1002 − 102

12 × 22

1002 × 102

82 ÷ 22

402 ÷ 52

(2 + 3)2

Evaluate: a

32 + 5

82 − 3 × 22

(7 − 3)2 + 10

2(32 + 1)2

Evaluate: a

13 + 23

43 + 23

33 – 23

53 – 33

43 × 23

73 × 43

93 ÷ 33

103 ÷ 53

(26 – 14)3

Evaluate: a

22 + 13

93 ÷ 32 – 13

(22 + 13)2

(33 – 2)2

(102 – 23)2 + 63

What would be the volume of a cubic aquarium with a side length of 22 cm?

Ace and his team arrange seating for a concert into a square grid with an equal number of rows and columns. If there are 18 chairs in each row, how many chairs will be needed in total?

Reasoning

14–15

14–16

16–19

The number 9 can be drawn as a square with side lengths of 3. What would happen if we tried to draw a non-square number as a square? Explain your answer.

Val was working out 103 − 73 and thought that he could simplify the expression using 10 − 7 = 3. a

Evaluate 103 − 73, by first expanding and evaluating each cube.

Evaluate 33.

Is 103 − 73 = (10 − 7)3 a true statement? Justify your answer.

2.02 Square and cube numbers mathspace.co

Lila was working out 23 × 33 and thought she could simplify it using 2 × 3 = 6. a

Evaluate 23 × 33 by first expanding and evaluating each cube.

Evaluate 63.

Is Lila correct in thinking that 23 × 33 can be simplified using 2 × 3 = 6? Justify your answer.

Calculate 43 × 23 using the above method.

Najah was working out 62 ÷ 22 and thought that she could simplify the expression using 6 ÷ 2 = 3. a

Evaluate 62 ÷ 22, by first expanding and evaluating each square.

Evaluate 32.

Is Najah correct in thinking that to solve 62 ÷ 22 she could use the fact that 6 ÷ 2 = 3? Justify your answer.

Calculate 122 ÷ 32 using the above method.

Francis was working out 42 + 52 and thought that he could simplify the expression using 4 + 5 = 9. a

Evaluate 42 + 52, by first expanding and evaluating each square.

Evaluate 92.

Is 42 + 52 = (4 + 5)2 a true statement? Justify your answer.

Explain why the square of an odd number is always odd.

Problem-solving 20–22

21–22, 24

22–26

What would be the area of a square having a perimeter of 36 cm?

Barney was considering the pattern of square numbers and thought he spotted a pattern when calculating the differences between consecutive square numbers. a

Calculate the differences between consecutive square numbers. i iii

12 − 02 32 − 22

ii iv

22 − 12 42 − 32

What pattern did Barney identify when he looked at the answers from part (a)?

Barney isn’t sure the pattern will continue. Explain whether you think it will or will not continue forever.

Mathspace WA Curriculum – Year 7 mathspace.co

Answer these questions: a

Complete the table. One column has been done for you. Number Cube of number Final digit of cube number

7 343

19 6859

Complete the second table. Number Cube of number Final digit of cube number

12 1728

State the pattern you observe in the ﬁnal digit row.

Use your observation to predict the final digit of the cube numbers from 21 to 30.

Will this pattern continue? Please justify your answer.

A number is multiplied by 8 and then added to 32. The result is a perfect square. What is the smallest positive integer value of this number?

Find the smallest perfect square number that can be expressed as the sum of two smaller perfect square numbers.

What is the smallest number that Amy could add to 1225 to make it a perfect cube?

A mathematician was asked about his house number. He replied: “My house number is the lowest two-digit number that is not prime. If reversed its digits and subtract it from the original number, you get a perfect cube. If you take the sum of its digits and square it, you get a perfect square that is also a two-digit number.” What is the mathematician’s house number?

2.02 Square and cube numbers mathspace.co

2.03 Square and cube roots After this lesson, you will be able to… • define what a square root and a cube root are. • understand the relationship between squares and square roots, and cubes and cube roots. • use the square root and cube root symbols and find the square root and cube root of a number. • solve problems involving square roots and cube roots

The square root A square root is a way of finding a number that can be multiplied by itself to get a certain number. For example, if we want to find the square root of 16, we are looking for a number that, when multiplied by itself, equals 16. That number is 4, because 4 × 4 equals 16. We write the square root of 16 as . Square roots are like the opposite of squaring a number. When we square a number, we multiply it by itself. When we take the square root of a number, we find a number that can be multiplied by itself to get the original number. We can call these opposite operations inverse operations. Not all numbers have whole number square roots. For example, the square root of 2 is a decimal number that goes on forever without repeating.

Example 1 Evaluate

Create a strategy Find the square root by ﬁnding the number that when multiplied by itself would give 121.

Apply the idea Express 121 as a product of prime factors

Find the square root

Example 2 Evaluate

Create a strategy Perform the operation inside the square root symbol then ﬁnd the square root of the result.

Mathspace WA Curriculum – Year 7 mathspace.co

Apply the idea Multiply 5 and 12 by itself

Evaluate the multiplication

Evaluate the addition

Find the square root

Reﬂect and check = 13 since 132 = 169.

Example 3 Between which two whole numbers would

be found?

Create a strategy

Apply the idea

Find the closest square numbers directly above and below 12, and then ﬁnd their square roots.

The closest square numbers on either side of 12 are 9 and 16.

These will be the closest whole number roots . either side of

So, we know that . and

will be between

will be found between the consecutive natural numbers 3 and 4.

Idea summary The square root ﬁnds the number that multiplies by itself to give a certain value. e.g. since 4 × 4 = 16.

The cube root If we are asked to find the cube root of a value, we are being asked, “What number multiplied by itself three times would give this value?” We might also see the cube root symbol written with a number inside it, for example, represents the cube root of 125 which is equivalent to 5 because 5 × 5 × 5 = 125. We can find the cube root of a perfect cube by thinking of a number that, when multiplied to itself three times, will equal the number underneath the cube root symbol. Another way to find the cube root is using the prime factorisation method. We can follow these steps: • Begin with the prime factorisation of the given number. • Partition the factors into groups of the same three factors. • Remove the cube root symbol and multiply the factors to get the answer.

2.03 Square and cube roots mathspace.co

Example 4 Evaluate

Create a strategy

Apply the idea

Think of a number that when multiplied by itself three times will equal to 27.

We know that 27 = 3 × 3 × 3. This means we get:

Example 5 What is

Create a strategy Use prime factorisation method and partition the factors into three groups of same factors to get the cube root. Multiply these factors.

Apply the idea

Prime factorisation

Partition factors into three groups of same factors

Evaluate the cube root

Evaluate the product

The cube root of 216 is 6.

Reﬂect and check Is 6 × 6 × 6 = 216?

Idea summary If we are asked to ﬁnd the cube root of a value, we are being asked, “What number multiplied by itself three times would give this value?”

Another way to ﬁnd the cube root is using prime factorisation method. We can follow these steps: • Begin with the prime factorisation of the given number. • Partition the factors into groups of three same factors. • Remove the cube root symbol and multiply the factors to get the answer.

Mathspace WA Curriculum – Year 7 mathspace.co

2.03 Practice questions Understanding 1

Explorer 1, 4, 5

Adventurer

Trailblazer

1–5

Complete: a b

64 = 8 × 8, therefore the square root of 64 is ⬚.

64 = 4 × 4 × 4, therefore the cube root of 64 is ⬚.

What is the relationship between the square root of a number and its square?

Some numbers have a whole number cube root. For example: Other numbers have a decimal cube root. For example: How can you identify whether a number has a whole number cube root?

Do these expressions result in whole numbers as their answers? a

Determine whether these statements are true or false: a

Fluency 6

8, 9c-hkl, 10, 12, 13c-f, 14, 15d-hkl, 17

6–7, 8cd, 9e-l, 11–12, 13e-h, 14, 15f-l, 16–17

Which has a value between 6 and 7?

Evaluate: a

Ex 2

A Ex 1

8, 9a-h, 10, 12, 13a-d, 15a-gl, 17

Which has a value between 10 and 11? A

Evaluate:

A square room has an area of 81 m2. Find the length of one side of the room.

Find the dimensions of a square plot of land of area 15 625 m2.

2.03 Square and cube roots mathspace.co

Consider the number a

Ex 3

Ex 4, 5

Complete the inequality with two consecutive perfect square numbers that 22 lies between: ⬚ < 22 < ⬚

Complete the inequality with two consecutive whole numbers that ⬚<

<⬚

Determine the two consecutive whole numbers between which each of these roots lie. a

Evaluate: a

lies between:

Evaluate:

State the integer that lies between

Sam has a cube-shaped storage container with a volume of 350 m3. What is the approximate length of each side of his container to the nearest metre?

Reasoning

and

18, 20

19–20, 22–23

Write a cube root that has a value between 4 and 5. Justify your answer.

Explain how you can estimate the value of a square root of a non-perfect square without a calculator. Give an example to support your answer.

Ansha claims that

is approximately equal to 6.

Identify and correct Ansha’s mistake. Explain your reasoning. 21

Donny is asked to simplify the expression

a 22

Identify the mistakes Donny made.

Show the correct working.

Consider the number a

Which pair of consecutive whole numbers does A

. His working is shown below:

9 and 10

10 and 11

4 and 5

Based on your answer in part (a), which number is

Mathspace WA Curriculum – Year 7 mathspace.co

lie between? D

3 and 4

closer to? Explain your reasoning.

Consider the number a

Which pair of consecutive whole numbers does A

3 and 4

4 and 5

Harper was working out the fact that 4 × 25 = 100. a

5 and 6

Evaluate

6 and 7

closer to? Explain your reasoning.

Based on your answer in part (a), which number is

Problem-solving 24–25 24

lie between?

24–25

24, 26–28

and thought that she could simplify the expression using

, by ﬁrst evaluating each square root.

Now, using the fact 4 × 25 = 100, evaluate

a true statement?

Calculate

using the above method.

Anja has purchased a Rubik’s cube with a volume of 343 cm3. Find the total perimeter of its faces.

Find the perimeter of a square kitchen floor with an area 160 000 cm2.

William has been creating some art patterns using tiles. Consider the pattern of tiles shown:

n =1

n =2

n =3

n =4

Complete the table. Shape number (n)

Total number of tiles (T ) b

Using the table in part (a), describe using words or symbols, the connection between the shape number (n) and the total number of tiles (T ).

Complete the table. Shape number (n)

Number of light purple border tiles (B) d

Using the table in part (c), describe using words or symbols, the connection between the shape number (n) and the number of light purple border tiles (B).

Using your answer from part (d), which shape would have 24 border tiles?

2.03 Square and cube roots mathspace.co

Using your answers from parts (b) and (d), determine the total number of tiles required if there are 36 border tiles.

Find the number of blue central tiles required when there are 100 tiles in total.

Find the number of border tiles required when there are 49 blue central tiles.

Letisha is digging a hole for a pond in her yard. The hole is currently in the shape of a cube and has a volume of 27 ft3. a

What is the length of one side of the hole?

How much wider do the sides need to be for the hole to be in the shape of a cube with a volume of 125 ft3?

2.04 Powers of whole numbers After this lesson, you will be able to… • define and understand the concept of powers of whole numbers. • write powers of whole numbers using index notation. • simplify and evaluate expressions using powers and indices.

Powers of whole numbers An index or power, is a number that indicates how many times a base number is multiplied by itself. The base number is usually a whole number (an integer) and the index is written as a small number in the upper right hand corner of the base: Index / Exponent / Power

103 = 10 × 10 × 10 Multiplied 3 times Base

The base number written with the small power to the top right is called index notation. The index tells us how many times the base number is multiplied by itself. When this is written out with all the multiplications shown, this is called expanded notation.

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The index is equal to the number of times the base is multiplied together. Some powers have special names. Squares, we have seen, with a power of 2. Cubes are numbers multiplied by themselves three times with a power of 3. For powers higher than 4, we say “to the power of “. 68

Mathspace WA Curriculum – Year 7 mathspace.co

Power 2 3

Example 52 53

Description 5 squared 5 cubed 5 to the power of 6

Expanded notation 5×5 5×5×5 5×5×5×5×5×5

Example 1 Write 5 × 5 × 5 × 5 × 5 × 5 using index notation.

Create a strategy Use the base and index deﬁnition: baseindex Count the number of times the base appears. This number will be the index or power when we write the expression in index form.

Apply the idea 5 is multiplied by itself 6 times. 5 × 5 × 5 × 5 × 5 × 5 = 56

The base is 5 and the index is 6

Idea summary 34 is a base number (3) raised to a power (4) and can be read as “Three to the power of four” and means 3 × 3 × 3 × 3 Index notation is what we call a number written using powers or indices. e.g. 211 Expanded notation is what we call a number written as a series of multiplications e.g. 2 × 2 × 2

Simplify or evaluate indices If we have an expanded expression like 3 × 3 × 3 × 3 × 3 × 3, how can we simplify it? When we simplify numbers in expanded notation, we rewrite the expression into index notation. We need to identify the base and index. In this case, the base is 3 and the index is how many times we multiply the base by itself. 3 × 3 × 3 × 3 × 3 × 3 = 36 If instead we are asked to evaluate 36, then we perform the expanded notation. 36 = 3 × 3 × 3 × 3 × 3 × 3 = 729 Evaluate means to find the value of.

2.04 Powers of whole numbers mathspace.co

Example 2 Complete the table of powers of 5. 51 5

Create a strategy Write the power terms in expanded notation and then perform the multiplication.

Apply the idea 52 = 5 × 5 = 25 3

5 =5×5×5 = 125 4

5 =5×5×5×5 = 625

Idea summary To simplify an expression in expanded notation, write it using index notation. This involves rewriting the expression with a base raised to a certain index. To evaluate the expression, perform the multiplication indicated by the expanded notation.

2.04 Practice questions Understanding

Explorer 1–4

State the base for the expression 95.

State the power for the expression 27.

For the pattern 101, 102, 103, 104, …, 108:

Trailblazer 1–4

How does the value of the expression change as the power increases?

How does the value of the expression change as the power decreases?

What repeated operation does an index represent? A

Adventurer 1–4

Addition

Subtraction

Mathspace WA Curriculum – Year 7 mathspace.co

Multiplication

Division

5–7, 8a-f, 9a-d, 10ab, 11–12

Fluency 5

Two squared

Three cubed

Ex 2

Two cubed

Expanded notation

Index notation

Write these powers of 9 in index form: 9×9

9×9×9

9×9×9×9

9×9×9×9×9

Write these expressions using index notation: a

5 × 5 × 5

1×1×1×1×1×1

3 × 3 × 3 × 3

2×2×2×2

4 × 4 × 4 × 4 × 4

7×7

12 × 12 × 9 × 9 × 9

11 × 7 × 11 × 7 × 7

12 × 12 × 12 × 15 × 15

2×4×2×4

24 × 24 × 24 × 12

2×2×3×3×3×5×5

Write these expressions in expanded form: a

46 × 92

775 × 132

Write these numbers in expanded form: a

Complete the table:

a 8

Five squared

Words Five to the power of six Six to the power of four Eight cubed Eleven to the power of three Twenty-three to the power of ﬁve Seven squared Fifteen to the power of four Ex 1

6, 8g-l, 10cd, 13

Write these statements using index notation: a

6, 8c-h, 9e-h, 10ac, 11–13

75 × 64

Consider the expression 53. a

Write the expression in expanded form.

Using your answer to part (a), evaluate 53.

Complete the table of powers of 2: 21

2 13

Complete these patterns: a

Powers of 3: 1, 3, ⬚, ⬚, 81, ⬚, ⬚

Powers of 5: 1, ⬚, 25, ⬚, 625 2.04 Powers of whole numbers mathspace.co

Reasoning

14–15

14–16

Explain how to evaluate 45.

Answer these: a

14–16

Complete the table: 11

1 b 16

Looking at the answers in the table, what could you say about any power of 1?

Answer these: a

In the table, what do you notice about the answers as you move from right to left? Now consider if an extra column was added to the start of the table.

What number would complete the table? c

Using a similar method, complete these tables. i

30 31 32 33 34 3

iii

100

125

625

101

102

103

104

100

1000

10 000

State a generalisation about any base that has a power of zero. 17a-f, 18e-h, 19–20

17, 18e-l, 19–20

Complete these statements: a

Problem-solving 17a-d, 18a-d, 19 17

024 = ⬚

3⬚ = 81

2⬚ = 32

5⬚ = 625 118

5⬚ = 125 2⬚ = 512

53 − 32

33 × 32

22 + 43

72 + 34

09 + 18 + 27

33 − 22 × 42

50 ⬚ 15

130 = ⬚

3⬚ = 729

Evaluate: a

23 × 42

122 − 22

53 − 33

State which symbol ( < , = or > ) would make these statements true: a

32 ⬚ 31

118 ⬚ 124

28 ⬚ 35

A number of bacteria triples in size every hour. If the initial amount of bacteria is 3, determine how many bacteria will be present after 3 hours.

Mathspace WA Curriculum – Year 7 mathspace.co

2.05 Powers of ten After this lesson, you will be able to… • identify the value of any power of ten. • rewrite numbers using powers of ten.

Powers of ten Recall that an index (or power) tells us the number of times to multiply the base by itself.

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The table shown demonstrates another way to think of some of the powers of ten that we see in the applet. Index form

Meaning

Value

105

10 × 10 × 10 × 10 × 10

100 000

One hundred thousand

104

10 × 10 × 10 × 10

10 000

Ten thousand

10 × 10 × 10

1000

One thousand

102

10 × 10

100

One hundred

Ten

One

10 10

In Words

Example 1 Complete this expression: 4000 = ⬚ × 103

Create a strategy

Apply the idea

Remember that multiplying by 10 is the same as multiplying by 1000. So, this question is asking us to ﬁnd the number that multiplies by 1000 to give 4000.

Think about how many times 1000 ﬁts into 4000. 4000 = 4 × 103

2.05 Powers of ten mathspace.co

Example 2 The distance between two stars is approximately 9 × 107 metres. Express this distance as a whole number.

Create a strategy

Apply the idea 7

Remember that multiplying by 10 is the same as multiplying by 10 000 000 (the number of zeroes is the same as the power).

Complete the multiplication 9 × 10 000 000. So, we have Distance = 90 000 000 metres

Idea summary When ﬁnding powers of ten, the answer will be equal to 1 followed by the same number of zeros as the power. For example, 102 = 100 because the power is 2, so there are two zeros following the 1 in the answer.

Rewrite numbers using powers of ten Since our place value system is based on powers of 10, all numbers can be written in expanded form using powers of 10. For example, consider the number 3842. To break it into powers of ten, we can look at each digit and its position in the number using a place value table: Thousand 1000 103 3

Hundreds 100 102 8

Tens 10 101 4

Ones 1 100 2

So, we can represent the number 3842 in the expanded form: 3842 = (3 × 1000) + (8 × 100) + (4 × 10) + (2 × 1) = 3000 + 800 + 40 + 2 Or with powers of ten: 3842 = 3 × 103 + 8 × 102 + 4 × 101 + 2 Make sure to use zero as a placeholder if there are no multiples of that power of ten in the number. For example: Number 4068 23 079

Expanded form 4000 + 60 + 8 20 000 + 3000 + 70 + 9

If we are given the number expressed in index notation, we can do the same process but in reverse, by evaluating the powers of ten to write the number in expanded form and then sum all of the terms.

Mathspace WA Curriculum – Year 7 mathspace.co

Example 3 Express 2487 in expanded notation using powers of ten.

Create a strategy Determine the place value for each digit and express it as a multiple of that place value.

Apply the idea 2487 = 2 × 1000 + 4 × 100 + 8 × 10 + 7 × 1 Expand the place value for each digit Rewrite each place value (power of ten) using = 2 × 103 + 4 × 102 + 8 × 101 + 7 expanded notation So, expressing 2487 in expanded notation using powers of ten, we have 2487 = 2 × 103 + 4 × 102 + 8 × 101 + 7

Example 4 Write the number expressed by: 2 × 105 + 5 × 104 + 6 × 103 + 7 × 101

Create a strategy Evaluate each of the terms with powers of ten.

Apply the idea We evaluate the powers of ten for each term: 2 × 105 + 5 × 104 + 6 × 103 + 7 × 101 = 2 × 100 000 + 5 × 10 000 + 6 × 1000 + 7 × 10 Evaluate each of the products: 2 × 100 000 + 5 × 10 000 + 6 × 1000 + 7 × 10 = 200 000 + 50 000 + 6000 + 70 Find the sum of all the terms: 200 000 + 50 000 + 6000 + 70 = 256 070 So, 2 × 105 + 5 × 104 + 6 × 103 + 7 × 101 = 256 070

Idea summary As our place value system is based on powers of 10, all numbers can be written in expanded form using powers of 10. For example: 3842 = 3000 + 800 + 40 + 2 = 3 × 103 + 8 × 102 + 4 × 101 + 2

2.05 Powers of ten mathspace.co

2.05 Practice questions Understanding

Adventurer

1–3

Trailblazer

1–3

1–2

How many zeros will 1012 have when written as a whole number?

105 is equal to: A

1 million

10 thousand

10 million

100 thousand

Which is smaller: 105 or 107? 4–5, 6a-d, 7a-d, 9a-f, 10a-c

Fluency

Ex 1

Explorer

4, 6aceg, 7c-f, 8, 9c-g, 10aceg

6aefh, 7e-h, 8, 9e-h, 10bdfh

Complete the table: 101

102

100

103

104

105

106

107

108

Complete the statement: 60 000 = ⬚ × 104

Write these as whole numbers: a

2 × 105

8 × 10

3 × 102

7 × 103

1 × 104

3×1

5 × 107

6 × 10

Express these numbers in expanded notation using powers of ten: a

100

1000

400

900

6000

7000

80 000

30 000

Ex 2

The distance between two planets is approximately 5 × 108 kilometres. Express the distance as a whole number.

Ex 3

Express these numbers in expanded notation using powers of ten:

Ex 4

2200

8300

2480

7020

6260

92 000

33 050

62 480

Write these as whole numbers: a

3 × 104 + 7 × 103

1 × 103 + 4 × 101

4 × 102 + 1 × 101

7 × 103 + 6 × 102 + 4 × 101

6 × 104 + 3 × 102 + 5 × 1

3 × 104 + 2 × 103 + 9 × 101 + 5 × 1

2 × 104 + 2 × 103 + 7 × 102 + 7 × 101 + 6 × 1

3 × 10 + 2 × 10 + 4 × 10 + 2 × 10

Mathspace WA Curriculum – Year 7 mathspace.co

Reasoning

11–12

11–14

If 10a > 10b what can we say about the values of a and b?

Which is smaller: 9 × 104 or 2 × 105? Justify your answer.

What is the difference between multiplying by 10 and raising to the power of 10? Use an example to support your answer.

Explain how powers of ten can be used to quickly compare the sizes of different numbers.

Problem-solving 15

15–16, 17a

15–17

The distance from Earth to the Moon is approximately 3 × 105 + 8 × 104 + 4 × 103 kilometres. Express the distance as a whole number.

The total mass of gold on Earth is approximately 1 × 108 + 7 × 107 + 1 × 106 + 3 × 105 kilograms. Express the mass as a whole number.

The distance in kilometres between the Sun and the first three planets is as follows: Planet Mercury Venus Earth

Distance from the Sun in kilometres 5 × 107 + 7 × 106 + 9 × 105 1 × 108 + 8 × 106 + 2 × 105 1 × 108 + 4 × 107 + 9 × 106 + 6 × 105

Find the distance between Mercury and Venus.

Find the distance between Venus and Earth.

Which is shorter, the distance between Mercury and Venus or the distance between Venus and Earth?

2.05 Powers of ten mathspace.co

2.06 Factors and multiples After this lesson, you will be able to… • find factors and multiples of a given number.

Factors A number is a factor if it divides another number with no remainder. For example, 4 is a factor of 12 because 12 ÷ 4 = 3 with no remainder. If we are finding all the factors of a number, we are trying to find all the numbers we can multiply together to make that specific number. Let’s find the factors of 12 by starting at 1 and trying to find a number that multiplies with it to make 12: 1 × 12 = 12 2 × 6 = 12 3 × 4 = 12 So, we can say that the factors of 12 are 1, 2, 3, 4, 6 and 12.

Example 1 Write down all the factors of 10.

Create a strategy We start at 1 and check each natural number to see if it is a divisor or factor of 10.

Apply the idea 1 × 10 = 10 2 × 5 = 10 5 × 2 = 10 10 × 1 = 10 The factors of 10 are: 1, 2, 5, 10.

Idea summary A factor is a number that divides another number evenly, leaving no remainder. Factors can be found by using multiplication tables. 4 is a factor of 12 because 12 ÷ 4 = 3 with no remainder.

Multiples A multiple is the product (multiplication) of a given number and any other integer. For example, 15 is a multiple of 5, because 15 can be written as 3 × 5 or thought of as “three lots of ﬁve”. 15 is also a multiple of 3 because it can be thought of as “ﬁve lots of three”. 78

Mathspace WA Curriculum – Year 7 mathspace.co

To find multiples of a number, we start by multiplying the given number by 1, then by 2, then by 3 and so on. To find multiples of 2, we work our way through these multiplications of 2: 2×1=2 2×2=4 2×3=6 So, the first three multiples of 2 are 2, 4 and 6. We can continue this process to find as many multiples as we like. We can also use multiplication tables, like the one shown, to find the first ten multiples of 2 by either reading across the row with the 2 on the left or down the column with a 2 at the top: x

100

Here are some other examples (we can use a multiplication table to find these too): • The first five multiples of 4 are 4, 8, 12, 16 and 20. • The first six multiples of 3 are 3, 6, 9, 12, 15 and 18. • The first five multiples of 7 are 7, 14, 21, 28 and 35.

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2.06 Factors and multiples mathspace.co

Example 2 Write the ﬁrst ﬁve multiples of 8.

Create a strategy

Apply the idea 8×1=8

Multiply 8 by the numbers 1, 2, 3, 4 and 5.

8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40 The ﬁrst ﬁve multiples of 8 are: 8, 16, 24, 32, 40.

Idea summary To find a specific multiple of a number, multiply that number by the position of the multiple you want. For example, if you want the third multiple of a number, multiply the number by 3.

2.06 Practice questions Understanding

Explorer 1, 3, 5

Adventurer 3–5

What is a factor of a number?

State the difference between a factor and a multiple of a number.

Which number is a factor of every number?

Which number is a factor of every even number?

How can you determine if a number is a multiple of another number?

Fluency 6

6–7, 8a-f, 9ab, 10a-d, 11–12, 14

Complete this table of factor pairs: Factor pairs of 50 (⬚, 50) (⬚, 25) (5, ⬚)

Mathspace WA Curriculum – Year 7 mathspace.co

7, 8e-j, 9bc, 10c-f, 11, 13–15

Trailblazer 2–3, 5

8g-l, 9cd, 10e-h, 13–15

Ex 1

Is 5 a factor of these numbers? a

155

252

Write all the factors of these numbers: a

231

312

Write all the factors of these numbers: a

Ex 2

115

Write down the ﬁrst ﬁve multiples of these numbers: a

Are these numbers multiples of 2? a

Write down the 8th multiple of 4.

Write down the 20th multiple of 4.

Fill in the blank: 42 is the ⬚th multiple of 7.

Lucy earns $30 a week. a

Fill in the table to help her keep track of her money.

How much money does Lucy have after 5 weeks?

Reasoning

18–20

17–20

Week 1 2 3 4 5 6

Total earnings (in dollars) 30 60

16–20

What types of numbers have exactly three factors? Justify your answer.

Can a multiple of a number also be a factor of that number? Explain your answer.

A teacher has 280 pencils that she wants to divide equally among her 35 students with no pencils left over. Determine if this is possible and explain your reasoning.

John has 40 marbles that he wants to divide equally among his 6 friends. a

How many marbles should each friend receive?

How many marbles are left over?

Explain why John has marbles left over.

2.06 Factors and multiples mathspace.co

Shirah has 16 ﬂowers to plant in her garden. She wants to arrange them in rows with the same number of ﬂowers in each row. a

State all the possible numbers of ﬂowers she can have in each row.

Explain how you determined your answer to part (a).

Problem-solving 21, 24–26

21–26

22–27

There are 12 eggs in a carton. If Harry buys 9 cartons for his shop, determine how many eggs he will have.

Three multiples of a certain number are 12, 15 and 27. The number is larger than 1. What is the number?

A company needs to pack boxes of toys and each box must have the same number of toys. If they have 144 toys, determine the largest number of boxes they can pack with no toys left over.

The company Mindt sells 6 deluxe bars of chocolate for $36. Madbury sells 4 bars of chocolate for $12. Which brand of chocolate is more expensive?

Two fishermen go fishing at different times during the week. Wally catches 12 fish per day 3 days a week whereas Callum catches 5 fish per day 4 days a week.

How many ﬁsh does Wally catch a week?

How many ﬁsh does Callum catch a week?

Who catches more ﬁsh a week?

A hotel has 180 rooms that need to be renovated. a

If the hotel wants to renovate the same number of rooms each week for 12 weeks, how many rooms should they renovate each week?

If they can only renovate 10 rooms a week, how many weeks will it take to renovate all the rooms?

Find the dimensions of a rectangular ﬁeld with a perimeter of 80 m and an area of 384 m2.

Investigation: The sieve of Eratosthenes

Investigate online

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2.07 Prime factorisations After this lesson, you will be able to… • express a given number as a product of its prime factors. • use a factor tree to perform prime factorisation.

Prime factorisation Every whole number greater than 1 is either prime or composite, with primes having only themselves and 1 as factors. 0 and 1 are special in that they are neither prime nor composite. Prime numbers are useful because they can be thought of as the building blocks of all other numbers (except 1). We can write any composite number as a product of prime numbers. One way to rewrite a composite number as a product of prime factors is using a factor tree. It is useful to know that the ﬁrst ﬁve prime numbers are 2, 3, 5, 7 and 11 for constructing factor trees. Start with the composite number we want to investigate, draw a box around it and draw two lines coming out of it. Here is how we might start with the number 12.

Next, choose two numbers that multiply together to make 12, such as 3 and 4. Write these at the end of each of the lines.

12 4

Circle any prime numbers. Since 3 is a prime number, we circle it. 4 is not a prime number, so we draw a box around it instead.

12 4

We then repeat the process with 4, with 2 × 2.

12 4 2

3 2

Since 2 is prime, we circle both of these numbers. Once all the numbers at the end of each branch are circled, the factor tree is complete.

12 4 2

3 2

Factor trees are not always unique - here is another factor tree for 12.

12 6 2

2 3

Multiplying together the circled numbers at the end of each branch always gives the original number. Here, we see 2 × 2 × 3 = 12.

We could have chosen 2 and 6 at the ﬁrst branch of the tree and still ended up with the same circled numbers 2, 2 and 3. These are called the prime factors.

2.07 Prime factorisations mathspace.co

Even if the numbers chosen at each branch are different, the numbers at the end of the branches will always be the same for any number — they will just be shown in a different order. So, the prime factors for every composite number are unique (there is only one set of possible prime factors). Here is a factor tree for 360. 360 30 15 3

12 2

4 2

We can write: 360 = 3 × 5 × 2 × 2 × 2 × 3

Multiply the prime factors together

360 = 2 × 2 × 2 × 3 × 3 × 5

Rewrite the factors in ascending order

360 = 2 × 3 × 5

Express the multiplications using index notation

Notice that the factor tree for 12 we made earlier is a smaller part of the factor tree for 360. This is because 12 is a factor of 360 and when we write 360 = 2 × 2 × 2 × 3 × 3 × 5 we can recognise the prime factorisation of 12 inside it: 360 = 2 × (2 × 2 × 3) × 3 × 5.

Interactive exploration

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Example 1 A number has this factor tree:

2 5

a What is this number at the top of the tree?

Create a strategy Multiply the prime factors together to obtain the composite number at the top of the factor tree.

Apply the idea Number = 2 × 2 × 5 × 5 × 7 × 11 = 7700

Mathspace WA Curriculum – Year 7 mathspace.co

Multiply the numbers at the end of each branch Perform the multiplication

b Write the prime factorisation of the number in index notation.

Create a strategy Rewrite the factors in ascending order and ﬁnd the certain prime factor that appears more than once and express it in index notation.

Apply the idea Number = 2 × 2 × 5 × 5 × 7 × 11 2

= 2 × 5 × 7 × 11

Multiply the numbers at the end of each branch Express the multiplications using index notation

Example 2 Write 148 as a product of prime factors in expanded form.

Create a strategy Start by dividing the number by the smallest possible prime number and repeating the same process with its quotient and so on until the quotient becomes a prime number.

Apply the idea 148 ÷ 2 = 74

Divide by 2

74 ÷ 2 = 37

Divide the quotient by 2

Since 37 is a prime number, we stop here. So we now have the expanded form: 148 = 2 × 2 × 37

Idea summary A number is a prime if it has exactly two factors: 1 and itself. A number is composite if it has more than two factors. Every composite number can be written as a product of its prime factors. For example, 30 = 2 × 3 × 5. Prime factors can be found using a factor tree. To construct a factor tree: • Start with a given number and branch out to two factors. • Any prime numbers are the end of a branch. • Each composite number in the factor tree branches out to two more factors until the end of every branch in the tree is a prime number. • These prime numbers are the prime factors of the original number.

2.07 Prime factorisations mathspace.co

2.07 Practice questions Understanding 1

1–2

Trailblazer

1–3

Prime number

iii

Prime factorisation of a number

A positive whole number that divides the number without leaving a remainder.

A positive whole number greater than 1 that has no positive integer divisors other than 1 and itself.

A unique representation of the number as a product of prime numbers.

Factor of a number

Identify which numbers are prime and which are composite in each list: 3, 10, 12, 4, 5, 6

14, 7, 15, 13, 11

21, 19, 27, 17, 1

How are factors and prime factors the same? How are they different?

Fluency 4

Adventurer

Match the terms to the deﬁnitions:

a 3

Explorer

4–5, 7a-d, 8a-d, 9a-d

4–6, 7c-f, 8c-f, 9c-f

4–6, 7e-h, 8e-h, 9e-h

The given ﬁgure shows a partial factor tree of 36.

Construct the remaining factors to complete the factor tree. 2

A number has this factor tree: What is the number at the top of the tree?

6 Ex 1

A number has this factor tree: a

What is the number at the top of the tree?

Write the prime factorisation of the number in index notation.

3 2

Mathspace WA Curriculum – Year 7 mathspace.co

5 3

Ex 2

For each number: i

Construct a prime factor tree.

Write the prime factorisation of the number in index notation.

126

144

200

360

Write these numbers as a product of their prime factors in expanded form: a

132

150

Write these values as a product of their prime factors using index notation: a

200

180

225

208

400

560

504

Reasoning

10–11

10–12

Can two different numbers have the same prime factorisation? Explain your reasoning.

Rae and Josie are working together to ﬁnd the prime factorisation for 30. Rae says they should begin by dividing by 5, but Josie says they should begin by dividing by 2. Who is correct? Explain your reasoning.

Explain how we can use prime factors to determine if 35 is a factor of 420 without using a calculator.

Problem-solving 13ace, 14ace 13

175 = 35 × 5

20 = 2 × 2 × 5

286 = 2 × 7 × 11

27 = 33

189 = 33 + 7

20 = 2 + 5

Do these equations represent the correct prime factorisation? If not, write the correct one using either expanded form or index notation. a

56 = 23 × 3

495 = 22 × 55

200 = 23 × 52

63 = 32 × 7

300 = 22 × 3 + 52

e 15

13bdf, 14bdf, 15–16

Do these equations show the correct prime factorisation in expanded form? If not, write the correct one.

e 14

13bdf, 14bdf, 15

504 = 2 + 3 + 7

What is the largest positive number that can evenly divide 108, 252 and 432 without leaving any remainder?

2.07 Prime factorisations mathspace.co

Think of any three-digit number. Write the number twice to make it a six-digit number. For example, if your number is 123, the new number will become 123 123. Then: • Divide your number by 7 • Divide your number by 11 • Divide your number by 13 a

Describe what happens after completing the process.

Repeat the process with other numbers. Based on your ﬁndings, what does this tell us about the prime factors of these types of six-digit numbers?

Write the product of 7, 11 and 13 in expanded notation using place value.

Use your ﬁndings to explain why the process above will work for any three-digit number.

Did you know?

Builders often use the concept of square roots without even realising it! Imagine tiling a large square patio. When you remove tiles from the patio in equal sections, you’re left with smaller squares that are perfectly proportional. This is much like finding the square root of a number, which is the process of breaking a large square into smaller identical parts. For example, if a patio is 25 square metres, the square root helps you determine that the side length of the patio must be 5 metres. Understanding square roots can make tasks like evenly dividing areas or materials much more efficient. Square roots are also essential in many other fields, from construction to navigation, helping to simplify large quantities into manageable, equal parts.

Investigation: Celtic knots

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Mathspace WA Curriculum – Year 7 mathspace.co

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2.08 HCF and LCM After this lesson, you will be able to… • determine the highest common factor of two or more numbers. • determine the lowest common multiple of two or more numbers.

Highest common factor (HCF) The highest common factor (HCF) is the largest factor shared by two or more given numbers. For example, the HCF of 12 and 18 is 6, which we can see: Factors 1, 2, 3, 4, 6, 12 1, 2, 3, 6, 9, 18

12 18

Comparing each list of factors, we can see that 6 is the largest factor shared by 12 and 18. To find the HCF for two numbers, you can simply look at the lists of factors for each number and find the highest number in common. This will always work, but there is a faster way using factor trees. To find the highest common factor of 126 and 294, we start by drawing their factor trees: Prime factorisation in expanded form: 126 = 2 × 3 × 3 × 7

126 6 2

21 3

Prime factorisation in expanded form: 294 = 2 × 3 × 7 × 7

294 6 2

49 3

We then ﬁnd what factors appear in both factorisations — they each have one 2, one 3 and one 7 in common. The highest common factor is the product of the common prime factors, 2 × 3 × 7 = 42.

2.08 HCF and LCM mathspace.co

Example 1 Find the highest common factor of 150 and 880.

Create a strategy Check the numbers that appear in both factorisations, then multiply these together.

Apply the idea List the prime factorisations ﬁrst: 150 = 2 × 3 × 5 × 5 880 = 2 × 2 × 2 × 2 × 5 × 11

Both factorisations contain one 2 and one 5, so we should multiply these together to ﬁnd the HCF. HCF = 2 × 5 = 10

Evaluate the product

Idea summary The highest common factor (HCF) is the highest factor that is common to all numbers. It can be found by listing factors or by writing each number as a product of prime factors and then multiplying the common prime factors together.

Lowest common multiple (LCM) Finding and using the LCM can help with efficiency in calculations involving fractions and ratios, but it isn’t always necessary. Sometimes using any common multiple is sufficient. This section looks at finding the LCM specifically. The lowest common factor (LCM) is the smallest multiple shared by two or more other numbers. For example, the LCM of 12 and 18 is 36, which we can see by writing down the ﬁrst four multiples of these numbers: 12 18

Multiples 12, 24, 36, 48, … 18, 36, 54, 72, …

Comparing each list of multiples, we can see that 36 is the ﬁrst multiple shared by 12 and 18, which makes it the lowest common multiple of these two numbers. To ﬁnd the LCM of two numbers, we can always create a list of multiples, but just like for HCF, there is a faster way using factor trees.

Mathspace WA Curriculum – Year 7 mathspace.co

To ﬁnd the lowest common multiple of 126 and 294, we look again at their factor trees and jump straight to the prime factorisations in expanded form, since we have found these previously: Prime factorisation in expanded form: 126 = 2 × 3 × 3 × 7

126 6 2

21 3

Prime factorisation in expanded form: 294 = 2 × 3 × 7 × 7

294 6 2

49 3

Find the HCF which would be 2 × 3 × 7 = 42. Next, we identify any factors that would be left over for each number after we remove the factors that make up the HCF. • For 126, after removing the common factors we are left with the factor 3. • For 294, after removing the common factors we are left with the factor 7. To ﬁnd the LCM, we multiply the HCF by these remaining values, so the LCM of 126 and 294 is 42 × 3 × 7 = 882.

Example 2 Consider these prime factorisations: 2200 = 2 × 2 × 2 × 5 × 5 × 11 2750 = 2 × 5 × 5 × 5 × 11 Find the lowest common multiple of 2200 and 2750.

Create a strategy Find the HCF of 2200 and 2750. Multiply the HCF by the prime factors left over from each number after removing the HCF factors.

Apply the idea HCF = 2 × 5 × 5 × 11 Find the HCF by identifying common factors = 550

Evaluate the product

Prime factors of 2200 not in the HCF = 2 × 2 Identify the remaining prime factors Prime factors of 2750 not in the HCF = 5 LCM = 550 × 2 × 2 × 5 Find the HCF by identifying common factors = 11 000

Evaluate the product

2.08 HCF and LCM mathspace.co

Idea summary The lowest common multiple (LCM) is the lowest multiple that is common to all numbers. It can be found by listing multiples or by finding the HCF and then multiplying that by the remaining factors.

Application of HCF and LCM There are many applications of HCF and LCM in real life. For example, we can use them when we want to make groups of objects, to know when an event will repeat or when two events will happen again at the same time. But, how can we tell if a word problem requires us to use HCF or LCM to solve it? When we are asked to find the HCF between two or more numbers, we are being asked what is the biggest number that they can all be divided by that leaves no remainder (or leftovers). When we are asked to find the LCM, we are being asked to find the smallest multiple that is shared by two numbers. We will use HCF for problems that require us to: • Split things into smaller groups or sections like in packs, in bags or in boxes • Equally share any quantity of items into their largest grouping • Arrange something into rows or columns or sets Some keywords that can help us identify if a problem requires finding HCF are: • Biggest, greatest, highest or maximum • Dividing, sharing, distributing or cutting into pieces We will use LCM to solve problems that ask us to: • Find when an event will be repeated or happen again • Find the number of pieces to collect multiple items in order to have enough Some keywords that can help us identify if a problem requires finding LCM are: • Smallest, least or minimum • Repeated over and over • Next

Example 3 Sophia has two pieces of cloth. One piece is 63 centimetres wide and the other piece is 42 centimetres wide. She wants to cut both pieces into strips so that all of the strips have the same width. a How wide should she cut the strips if she wants the strips to be as large as possible?

Create a strategy

Apply the idea

Look for the largest number that will divide both widths without a remainder.

Out of the shared factors of 63 and 42, the highest is 21.

Look for the highest common factor of 63 and 42.

So, the largest width the strips could have is 21 centimetres.

Mathspace WA Curriculum – Year 7 mathspace.co

b How many strips will she end up with?

Create a strategy We can find the number of strips made from the first piece of cloth by dividing 63 by 21 and from the second piece by dividing 42 by 21. Add these two results to find the total number of strips.

Apply the idea Number of strips = (63 ÷ 21) + (42 ÷ 21) =3+2

Evaluate the division

Evaluate

Idea summary The concept of HCF is frequently applied in daily life, such as when evenly dividing two or more groups of items into the largest grouping possible. The LCM can be used to ﬁnd the smallest multiple of two or more numbers and solve a range of related problems, including ﬁnding the time it takes for multiple events to occur together.

2.08 Practice questions Understanding 1

Adventurer 1–2

Trailblazer 1–2

Deﬁne: a

Explorer 1–2

Highest common factor (HCF)

Lowest common multiple (LCM)

Are these statements true or false? Correct the statement if it is false. a

The HCF of three consecutive even numbers is always 2.

If the HCF and LCM of two numbers are equal, then the numbers are distinct.

The HCF of two prime numbers is always 1 because the only common factor of two prime numbers is 1.

The HCF of two numbers cannot be smaller than either of the two numbers because a factor of a number cannot be smaller than the number itself.

The LCM of two numbers cannot be less than either of the two numbers because a multiple of a number must be less than or equal to the number itself.

2.08 HCF and LCM mathspace.co

Fluency 3

3–6, 7a-d, 9a, 10–11, 12a-d, 15–16

3–5, 7e-h, 8, 9ab, 11, 12c-f, 13–17

6, 7g-j, 8, 9cd, 11, 12e-h, 13–17

For each of these numbers: i

List the factors.

6 and 12

9 and 24

State the highest common factor.

14 and 32

28 and 42

Given these prime factorisations: 180 = 2 × 2 × 3 × 3 × 5 600 = 2 × 2 × 2 × 3 × 5 × 5 Find the highest common factor of 180 and 600.

Given these prime factorisations: 224 = 2 × 2 × 2 × 2 × 2 × 7 196 = 2 × 2 × 7 × 7 Find the highest common factor of 224 and 196.

Given these prime factorisations: 4900 = 2 × 2 × 5 × 5 × 7 × 7 1750 = 2 × 5 × 5 × 5 × 7 Find the highest common factor of 4900 and 1750.

7 Ex 1

Find the highest common factor of these groups of numbers: a

8 and 16

18 and 24

21 and 33

36 and 48

42 and 24

72 and 108

121 and 143

150 and 560

48, 80 and 176

168, 312 and 120

The table shows the prime factorisations of several numbers in index form. a b

What is the highest common factor of 2940 and 14 700? What is the highest common factor of 17 640 and 308 700?

Number

Prime factorisation

2940

22 × 3 × 5 × 72

14 700

22 × 3 × 52 × 72

17 640

23 × 32 × 5 × 72

26 250

2 × 3 × 54 × 7

308 700

22 × 32 × 52 × 73

For each of these numbers: i

List the ﬁrst 8 multiples.

6 and 10

8 and 14

State the lowest common multiple.

24 and 42

Given these prime factorisations: 54 = 2 × 3 × 3 × 3 36 = 2 × 2 × 3 × 3 Find the lowest common multiple of 54 and 36.

Mathspace WA Curriculum – Year 7 mathspace.co

15 and 30

Given these prime factorisations:

Ex 2

1260 = 2 × 2 × 3 × 3 × 5 × 7 1890 = 2 × 3 × 3 × 3 × 5 × 7 Find the lowest common multiple of 1260 and 1890.

Find the lowest common multiple of these groups of numbers: a

6 and 14

7 and 18

24 and 84

78 and 26

144 and 156

450 and 650

3, 8 and 18

9, 14 and 27

Consider the numbers 1575 and 1650. a

What is the highest common factor of 1575 and 1650?

What is the lowest common multiple of 1575 and 1650?

The table shows the prime factorisations of several numbers in index form. a b

15 Ex 3

What is the lowest common multiple of 2940 and 17 640? What is the highest common factor of 14 700 and 26 250?

Number

Prime factorisation

2940

22 × 3 × 5 × 72

14 700

22 × 3 × 52 × 72

17 640

23 × 32 × 5 × 72

26 250

2 × 3 × 54 × 7

308 700

22 × 32 × 52 × 73

Allena has two pieces of cloth. One piece is 68 centimetres wide and the other piece is 32 centimetres wide. She wants to cut both pieces into strips so that all of the strips have the same width. a

How wide should she cut the strips if she wants the strips to be as large as possible?

How many strips will she end up with?

Cicada species spend many years underground, emerging as a single brood every generation. Platypedia areolata emerges every 19 years. They are hunted by the Argiope aurantia spider, which has a 4 year breeding cycle. a

How often do these species start their breeding cycle at the same time?

A genetic mutation changes the life cycle of the cicadas so the mutants emerge every 15 years. How often will the mutant cicada species start their life cycle at the same time as the spiders?

Two airlines, JetFuel and LionAir, each land planes at an airport terminal. During the day a JetFuel plane lands every 18 minutes and a LionAir plane lands every 16 minutes. A shuttle bus takes passengers from the terminal to the city centre, departing every 21 minutes. a

How often does a JetFuel plane land and a shuttle bus depart at the same time?

Which of these two options happens more often in a single day? A

JetFuel plane lands and the shuttle bus departs at the same time.

LionAir plane lands and the shuttle bus departs at the same time.

2.08 HCF and LCM mathspace.co

Reasoning

19–20

18–20

A student claims that two numbers will always have a common factor and a common multiple. Explain if the student is correct.

Explain how the prime factors of two numbers can be used to ﬁnd their highest common factor.

The HCF of two numbers is 4 and the LCM is 48. a

Sasha thinks the two numbers are 8 and 12. Determine if Sasha is correct. Explain your reasoning.

Suri claims the two numbers will be multiples of 4. Determine if Suri is correct. Explain your reasoning.

What could the two numbers be?

Problem-solving 21, 22

22, 23

21–24

Jane has a piece of cloth that has a size of 84 cm by 112 cm. She wants to cut squares of equal size out for a patchwork quilt without wasting any material. What would be the side length of the largest size square that can be cut?

Cicadas are a type of insect that spend most of their life underground and come above ground when they mature. Two variations of the species, Type A and Type B, emerge above ground as a group every 13 and 17 years respectively. If the last year that both Type A and Type B emerged was 1993, what is the next year that both types will emerge together?

Cheryl and Berting are juggling. They have 4 red balls and 1 green ball each. The time between releasing the green ball and releasing the green ball again is 6 seconds for Cheryl and 9 seconds for Berting. At some point in the performance, Cheryl and Berting each release their green ball at exactly the same time.

How long will it be until they each release their green ball at the same time again?

How many red balls has Berting thrown before they each release their green ball at the same time again?

Two students are training by running laps in an oval. They pace themselves and jog at constant, but different speeds. The faster student completes each lap in 90 seconds and the slower student completes each lap in 120 seconds. If both students begin at the starting line at the same time, how many minutes will pass before they both pass the start line at the same time again?

Mathspace WA Curriculum – Year 7 mathspace.co

2 Chapter review 1

Fill in the boxes to complete the working out using the associative law. 45 + 27 + 18 = 45 + (⬚ + ⬚)

32 + 52 − 24 = 32 + (⬚ − ⬚)

152

182

262

802

1202 + 102

62 − 32

122 − 82

162 − 52

(7 − 3)2 + 16

3 (42 + 1)2

a 2

Evaluate:

Evaluate: a

10 – 3

8 +6

(5 + 2)3 – 10

2(33 – 4)

15 ÷ 3

123

23 × 3

Jenny and her crew organise chairs for a school play in a square formation with the same number of chairs in each row and column. If there are 25 chairs in each column, how many chairs are required in total?

A man said to his friend: “My age in years is an even number that is not prime. If you reverse the digits, you get a perfect square. If you divide this perfect square by 4, you also get a perfect square.” How old is the man?

A librarian said to a visitor: “My age in years is an odd number that is not prime. If you take the cube root of my age and round to the nearest integer, you get an even number. If you subtract 1 from this even number, you get the cube root of a two-digit number.” How old is the librarian if he adds that he is less than 65 years old?

Evaluate:

Chapter 2 review mathspace.co

Consider the number a

Complete the inequality with two consecutive perfect square numbers that 70 lies between: ⬚ < 70 < ⬚

Complete the inequality with two consecutive whole numbers that ⬚<

lies between:

<⬚

Determine the two consecutive whole numbers each of these roots lies between. b

Jason has a square garden with an area of 81 m2. Find the perimeter of the garden.

Find the perimeter of a square auditorium with an area of 1 000 000 m2.

Leigh has purchased a sugar block with a volume of 512 cm3. Find the total perimeter of its faces.

Rewrite these expressions using index notation:

6 × 6 × 6 × 6

15 × 7 × 15 × 7 × 7

16 × 16 × 16 × 20 × 20

2×2×3×3×3×5×5

Write these expressions in expanded form: a

75 × 64

46 × 92

774 × 132

1205 × 122

Consider the expression 124. a

Write the expression in expanded form.

Using your answer to part (a), find the value of 124.

Evaluate: a

123 − 45

63 − 32

33 × 42

09 + 38 + 27

93 − 22 × 43

A rabbit population doubles in size every month. If the initial amount of rabbits is 12, determine how many rabbits will be present after 4 months.

Write these as whole numbers: a

8 × 105

7 × 106

9 × 107

69 000

Express these numbers in expanded notation using powers of ten: a

3 × 105

2200

8400

Write these as whole numbers: a

5 × 102 + 3 × 101

8 × 103 + 6 × 102 + 2 × 101

3 × 104 + 2 × 103 + 7 × 101 + 3 × 1

9 × 104 + 3 × 103 + 8 × 102 + 7 × 101 + 4 × 1

Mathspace WA Curriculum – Year 7 mathspace.co

50 000

A company sold 2 × 103 + 3 × 102 + 5 × 101 units of Product A and 4 × 102 + 7 × 101 + 6 units of Product B last year. Express the total number of units sold for both products as a whole number.

List all the factors of these numbers: a

Write down the 30th multiple of 2.

Write these values as a product of prime factors in expanded form: a

120

150

Given these prime factorisations: 24 = 2 × 2 × 2 × 3 144 = 2 × 2 × 2 × 2 × 3 × 3 Find the highest common factor of 24 and 144.

Consider the numbers 270 and 3465. a

What is the highest common factor of 270 and 3465?

What is the lowest common multiple of 270 and 3465?

The table shows the prime factorisations of several numbers in index form. a

What is the highest common factor of 1260 and 8820?

What is the highest common factor of 26 250 and 220 500? Number 1260 8 820 26 250 31 500 220 500

Prime factorisation 22 × 32 × 5 × 7 22 × 32 × 5 × 72 2 × 3 × 54 × 7 22 × 32 × 53 × 7 22 × 32 × 53 × 72

Sarah has a piece of fabric that has a size of 90 cm by 150 cm. She wants to cut squares of equal size out for a patchwork quilt without wasting any material. What would be the side length of the largest size square that can be cut?

Chapter 2 review mathspace.co

Big ideas The number line serves as a model to relate integers to statements of order in real-world contexts.

Curriculum Outcomes • Add and subtract integers using flexible and efficient strategies • Draw and label, or use a given number line, to locate, order and compare with equality and inequality symbols, fractions, terminating decimals, percentages and integers • Explore to extend addition and subtraction of positive integers to include negative integers

3 Integers Chapter outline 3.01 Integers and the number line 3.02 Compare integers Investigation: Life-size number line 3.03 Add and subtract integers Chapter 3 review

102 109 114 122

Negative numbers help meteorologists predict chilly temperatures!

3.01 Integers and the number line After this lesson, you will be able to… • plot integers on the number line. • determine the location of points on the number line. • identify opposite integers.

Integers on a number line Negative

Positive

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1

2 3 4 5 6 7 8 9 10

The integers are made up of positive and negative whole numbers, as well as the number 0. {…, −3, −2, −1, 0, 1, 2, 3, …} The integers to the right of 0 are positive integers and the integers to the left of 0 are negative integers. 0 is neutral; it is neither negative nor positive. The arrows at each end indicate that the line extends inﬁnitely in the positive and negative directions.

−5 −4 −3 −2 −1 0

2 3 4 5

On this number line, we can see that the point is at the integer −3. This point is 3 units to the left of 0.

−6

−3

On this number line, the location of the point has not been labelled. We can see that there are three gaps between the labels of 0 and 3. This means each gap will be equal to 1 unit, so the location of the point is 4. This point is 4 units to the right of 0.

Interactive exploration

Discover this concept in action online

102

Mathspace WA Curriculum – Year 7 mathspace.co

mathspace.co

We can also use the number line to ﬁnd the location of a point after it has been moved. If a point located at 1 is then moved 3 units to the right, it will end up at 4. −5 −4 −3 −2 −1 0

If the point at 1 is moved 3 units to the left, it will end up at −2. −5 −4 −3 −2 −1 0

Moving the point to the left will decrease the integer value and moving the point to the right will increase the integer value. Moving to the left is called moving in the negative direction. Moving to the right is called moving in the positive direction. If we move the point at 0 by 4 places in the positive direction, we get 4. −5 −4 −3 −2 −1 0

If we move the point at 0 by 4 places in the negative direction, we get −4. −5 −4 −3 −2 −1 0

Example 1 Where is the point plotted on the number line? −12

−8

−4

Create a strategy

Apply the idea

Since there are 4 units between every second tick, each tick is 2 units apart.

The point is two units to the left of −4. Number = −6

Example 2 A point is plotted on the number line. Find the number that is 7 units to the left of this plotted point. −10

−5

Create a strategy Find the number represented by the point, then move 7 steps to the left from the point.

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103

Apply the idea The plotted point is 3 steps to the right of 5. So the plotted point is 5 + 3 = 8. Moving 7 steps to the left from 8 will end up at 1.

−10

−5

Number = 1

Idea summary Integers are positive and negative whole numbers, as well as the number 0. Positive integers are to the right of 0. They are {1, 2, 3, …}. Negative integers are to the left of 0. They are {…, −3, −2, −1}. 0 is neutral; it is neither negative nor positive. Moving right on the number line can be called moving in a positive direction. This results in an increase in value. Moving left on the number line can be called moving in a negative direction. This results in a decrease in value.

Opposite integers Two integers are opposite integers if their locations on the number line are the same distance from 0, but on different sides of 0. This example shows that the opposite of 3 is −3. −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

We can also say −3 is the opposite of 3. −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

The opposite of the integer 0 is 0.

Example 3 Consider the number plotted on the number line. −18 −15 −12

−9

−6

−3

a State the number that has been plotted.

Create a strategy From 9, count up until reaching the plotted point.

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Apply the idea The plotted point is at the ﬁrst tick mark after 9. So the plotted point is 9 + 1 = 10.

b What number is the opposite of the number from part (a)?

Create a strategy Move 10 steps from 0 in negative direction.

Apply the idea Moving 10 steps in the negative direction from 0 will end up at −10.

−18 −15 −12

−9

−6

−3

Number = −10

Idea summary Opposite integers are located the same distance from 0, but on different sides of 0. The opposite of a positive integer is a negative integer. The opposite of a negative integer is a positive integer. The opposite of zero is zero.

3.01 Practice questions Understanding 1

Explorer 1–4

Adventurer 1, 3–5

Trailblazer 1, 4, 5

Determine whether each number is an integer or non-integer. a

−8

7.5

−9.66

1104

−1.400

Consider the integer 0 on the number line. a

Where are the positive integers located?

Where are the negative integers located?

Consider the integers on the number line: a

List the ﬁrst three positive integers.

List the ﬁrst three negative integers.

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105

0 is an integer. Is it negative, positive or neither?

Explain what opposite integers are; give an example. 6a-e, 7a-d, 8, 10, 12, 14ac, 15, 16a-d, 17

Fluency Ex 1

−4 −3 −2 −1

−3

−10

−5

−8

−4

−8

−4

−5

−8

−10

−14

Identify the integer that is 6 units to the right of the plotted point: −5

Identify the integer that is 9 units to the left of the plotted point: −5

Identify the integer that is 6 units to the left of the plotted point: −5

Identify the integer that is 7 units to the right of the plotted point: −5

Identify the integer that is 8 units to the left of the plotted point: −8

Plot these integers on a number line:

−10

106

h 3

−10

Ex 2

−4

−4 −3 −2 −1

−10

−8

−6

−5

6f-j, 7e-h, 11–14, 15cd, 16e-h, 18

Identify the integer where the point is plotted on the following number lines: a

6c-g, 7a-d, 9, 10, 12, 13, 14bce, 16a-d, 17, 18

−6

Mathspace WA Curriculum – Year 7 mathspace.co

−4

−2

Identify the integer that is 12 units to the right of the plotted point: −16

Ex 3

−8

−12

−4

Identify the integers that represent these statements: a

Two units to the right of 7

Three units to the left of 12

Two units to the left of −6

Four units to the right of −9

Three units left of 2

Ten units right of −3

For each of the following number lines: i

Identify the integer that has been plotted.

Identify the opposite of the integer that has been plotted.

a −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

b −14 −12 −10 −8 −6 −4 −2

−20

c −16

−12

−8

−4

d −21 −18 −15 −12 −9 −6 −3

Identify the integer that is the opposite of the following: a

−8

−5

−10

402

−88

123

Consider the integer −4. a

Plot −4 on a number line.

On the same number line, plot its opposite.

The opposite of an integer is −16. Identify the original integer.

Reasoning

20, 22

20, 22–24

19–21, 23, 24

True or False: “The opposite of a number is always greater than the number itself”. Explain your answer.

Explain the importance of 0 on the number line.

Explain why the opposite of 0 is 0.

Explain why the statements “cycling 8 km due south” and “cycling 8 km due north” can be thought of as opposite integer statements.

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The invention of the integer zero or the concept of nothingness as a number in around 500 AD, was a signiﬁcant development in the history of mathematics. Why do you think zero wasn’t invented earlier?

The number line stretches to positive and negative inﬁnity. Is inﬁnity an integer? Explain your answer.

Problem-solving 25 25

Identify the integer that would represent each situation: i

108

27–29

A gain of $400 can be represented by the integer 400. a

26–28

Gaining another $400

Losing $400

Identify the integer that represents your answer in part (a).

Consider the following statement: “Arriving 14 minutes late.” a

Identify the opposite of “Arriving 14 minutes late”.

Suppose “Arriving 14 minutes late” is represented by the number 14. Identify the integer that would represent its opposite.

Construct a statement and identify the integer that represents the opposite of each: a

“Descending 2 ﬂoors” is represented by the number −2.

“A weight gain of 11 kg” is represented by the number 11.

“A temperature of 10 °C below freezing point” is represented by the integer −10.

A temperature drop of 21 °C can be represented by the integer −21. a

Identify the integer that represents the opposite of a temperature drop of 21 °C.

Identify the integer that represents the opposite of the opposite of a temperature drop of 21 °C.

A mountain peak is measured to have a height of 5100 m above sea level. An ocean trench is found to be the same distance below sea level. a

If the height above sea level of the mountain peak is represented by the number 5100, identify the integer that represents the distance below sea level of the trench.

In this scenario, what does the integer 0 represent?

Mathspace WA Curriculum – Year 7 mathspace.co

3.02 Compare integers After this lesson, you will be able to… • compare integers. • order integers. • order and compare integers in real life situations.

Compare and order integers We can use the following 4 inequality symbols to compare integers: > means “is greater than”, such as 7 > 2 < means “is less than”, such as −2 < 4 ≥ means “is greater than or equal to” ≤ means “is less than or equal to” The number that is furthest right on the number line is always greater. −8 −7 −6 −5 −4 −3 −2 −1 0

Of the plotted points above: −4 is the furthest to the left; therefore it is the smallest integer marked on this number line. 8 is the furthest to the right; therefore it is the largest integer marked on this number line. Writing the numbers from smallest to largest is called ascending order, −4 < 0 < 3 < 8. Writing the numbers from largest to smallest is called descending order, 8 > 3 > 0 > −4.

Interactive exploration

Discover this concept in action online

mathspace.co

Example 1 Consider the statement 7 > −5. a What is the meaning of this inequality? A

7 is greater than 5

−5 is greater than 7

7 is greater than −5

Create a strategy

Apply the idea

Translate the inequality symbol > into “is greater The statement 7 > −5 can be translated into “7 is than”. greater than −5”. The answer is option C.

3.02 Compare integers mathspace.co

109

b Is this inequality true or false?

Create a strategy Plot the numbers onto the number line to compare their values.

Apply the idea Here are 7 and −5 on the number line. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Since 7 is to the right of −5, it shows that 7 is greater than −5. The inequality is true.

Example 2 Insert an inequality sign, < or > , between the two given numbers to make the statement true.

Create a strategy

−9 ⬚ 9

Plot the numbers onto the number line to compare their values.

Apply the idea Here are −9 and 9 on the number line. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Since −9 is to the left of 9, it shows that −9 is less than 9. The inequality symbol to be used is <. −9 < 9

Example 3 The melting point of krypton is −157 °C. The melting point of radon is −71 °C. Write an inequality comparing the two melting points.

Create a strategy

Apply the idea

A negative integer with a smaller number has a greater value than a negative integer with a larger number.

−71 is a negative integer with the smaller number which means it has a greater value than −157. −157 °C < − 71 °C

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Reflect and check −157 °C is to the left of − 71 °C. −160°C −150°C −140°C −130°C −120°C

−110°C −100°C

−90°C

−80°C

−70°C

Idea summary Use inequality symbols to compare integers. < means “less than”. For example, 7 < 11 is read as 7 is less than 11. > means “greater than”. For example, 3 > 2 is read as 3 is greater than 2. The further right a number is on the number line, the greater it is. Ascending order means going up in numerical order. For example: −2, 4, 7. Descending order means going down in numerical order. For example: 12, −4, −7.

3.02 Practice questions Explorer

Understanding 1

Adventurer

1–5

Consider the integers 6 and −4 plotted on the number line: −8 −7 −6 −5 −4 −3 −2 −1 0

Trailblazer 2–5

1–5

Which number is further to the right?

Which number is further to the left?

Which number is bigger?

Consider the integers −6 and −4 plotted on the number line: −10

−9

−8

−7

−6

−5

Which number is further to the right?

Which number is bigger?

−4

−3

−2

−1

Which number is further to the left?

If you have two integers plotted on a number line, how do you know which number is larger?

What is the meaning of the symbol, “ > ”?

What is the meaning of the symbol, “ < ”?

Explain the meaning of the phrase “ascending order”.

Explain the meaning of the phrase “descending order”.

Are the integers −4, −2, 0, 3, 6, 12 in ascending order or descending order?

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111

6–10, 11ab, 12a-d, 13a, 14a

Fluency 6

6b, 7b, 8cd, 9cd, 11cd, 12a-d, 13a, 14a

6b, 7b, 8cd, 9cd, 11c-f, 12e-h, 13b, 14b

For each number line: i

Arrange all the plotted integers in ascending order.

Identify the largest number.

a −10

−5

b −20

−16

−12

−8

−4

For each number line: i

Arrange all the plotted integers in descending order.

Identify the smallest number.

a −20

−15

−10

−5

b −12 −10

Ex 1

Ex 2

13, −14, 2

−7, −17, −12

8, −8, 7

35, −36, −40

−14, −4, −21

−7, −24, 10

1, −1, 5

34, −36, −2

8 is less than 10

2 is greater than −4

−5 is less than −1

−3 is greater than −8

For each of the following inequalities i

Write the inequality in words.

Determine whether the inequality is true or false

−7 < −2

8 > −6

7≥4

3≤2

0 > −1

23 < −47

Identify the inequality sign, < or > , that should appear between each pair of numbers to make each statement true: 7 ⬚ −4

39 ⬚ −72

b f

−8 ⬚ 8

−12 ⬚ −14

c g

0 ⬚ −8

−45 ⬚ 22

Write a statement using < or > that compares the two temperatures. a

112

e 13

−2

The statement “7 is greater than 4” can be written as 7 > 4. Write each mathematical statement using an inequality sign:

Ex 3

−4

Identify the smallest number in each set: a

−6

Identify the largest number in each set: a

−8

−19 °C and 4 °C

Mathspace WA Curriculum – Year 7 mathspace.co

−3 °C and −17 °C

d h

−2 ⬚ −5

−1 ⬚ −30

Write a statement using < or > that compares the two integers. a

−15 and −2

Reasoning

15, 17

−10 and 0

16, 17

Gian and Kim were asked whether −7 or −5 was the larger integer. Gian said −5 while Kim said −7 because 7 is the bigger number. Explain which person is correct and how you worked out the answer.

Before computers, accountants used red ink to record losses and black ink to record proﬁts. Even today, the saying “$8000 in the red” means an $8000 loss.

Explain what “$8000 in the black” means.

If it were your business, would you rather be $784 in the red or $225 in the black? Explain why.

The boiling point of the element krypton is −153 °C. The boiling point of the element radon is −62 °C. Which of these elements has the lower boiling point? Explain how you worked out your answer.

Problem-solving 18–20, 21abce 18

18–20, 21abce

18–20, 21bdf

Arrange these temperatures in order from coldest to hottest: −15 °C, 30 °C, −11 °C, 2 °C, −9 °C

The numbers −3, 4 and 2 can be written in ascending order using inequality notation like this: −3 < 2 < 4. Arrange this list of numbers in ascending order using inequality notation. 10, −25, 16, −11, 27

The numbers −3, 4 and 2 can be written in descending order using inequality notation like this: 4 > 2 > −3. Arrange this list of numbers in descending order using inequality notation. 3, −13, 11, −9, 15

Complete each number pattern: a c e

2, 4, 6, ⬚, ⬚, ⬚

−16, −12, −8, ⬚, ⬚, ⬚

6, 3, 0, ⬚, ⬚, ⬚

b d f

2, 1, 0, ⬚, ⬚, ⬚

−4, −1, 2, ⬚, ⬚, ⬚

−17, −12, −7, ⬚, ⬚, ⬚

Investigation: Life-size number line

Investigate online

mathspace.co

3.02 Compare integers mathspace.co

113

3.03 Add and subtract integers After this lesson, you will be able to… • add and subtract positive and negative integers using the number line. • solve addition and subtraction problems involving positive and negative integers.

Addition and subtraction using a number line The number line can help us to add and subtract integers. Both +3 and −3 have a magnitude of 3. +3 would be 3 jumps to the right on the number line. −3 would be 3 jumps to the left on the number line. The magnitude of an integer is how far that number is away from 0.

Interactive exploration

Discover this concept in action online

mathspace.co

Add a positive number Adding positive numbers will mean from whatever your starting point is, you will move right on the number line.

Example 1 Calculate −1 + 4.

Create a strategy Start on the number −1 and move 4 spaces to the right.

Apply the idea When we add a positive number, we move to the right. −1 + 4 = 3

114

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Mathspace WA Curriculum – Year 7 mathspace.co

Subtract a positive number Subtracting a positive number will mean from whatever your starting point is, you will move left on the number line. “Negative is the Opposite to Positive” We can think of the negative sign as meaning “opposite”. The negative sign causes us to move in the opposite direction.

Example 2 Calculate −1 − 4.

Create a strategy Start on the number −1 and move 4 spaces to the left.

Apply the idea When we subtract a positive number, we move to the left. −1 − 4 = −5

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Reflect and check We can think of the negative sign as meaning “opposite”, −5 is the opposite of 5. The negative sign causes us to move in the opposite direction.

Add a negative number Adding a negative number will mean from whatever your starting point is, you will move left on the number line. Adding a negative number is the same as subtracting that number.

Example 3 Calculate 4 + (−3).

Create a strategy Start on the number 4 and move 3 spaces to the left.

Apply the idea If we want to calculate 4 + (−3) we are adding a number, but because it is negative, we move left instead of right. 4 + (−3) = 1

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

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115

Subtract a negative number Subtracting a negative number means that from whatever your starting position is, you will move right on the number line. Subtracting a negative, becomes a positive.

Example 4 Calculate 4 − (−3).

Create a strategy Start at 4 and move 3 spaces to the right.

Apply the idea If we want to calculate 4 − (−3) we are subtracting a number, but because it is negative, we move right instead of left. 4 − (−3) = 7

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Reflect and check Why do we move right? If we added −3, we would move left, so if we subtract −3, it must mean to do the opposite, so we move right. We can see it looks like 4 + 3 so subtracting a negative is the same as addition.

Idea summary When adding or subtracting, the sign tells us the direction to move and the magnitude of the number tells us how far to move. To add and subtract using a number line: • Adding a positive integer means we move to the right, −3 + 5 means start at −3 and move 5 right. • Adding a negative integer means we move to the left, 3 + (−6) means start at 3 and move 6 left. • Subtracting a positive integer means we move to the left, −5 − 7 means start at −5 and move 7 left. • Subtracting a negative integer means we move to the right, 2 − (−8) means start at 2 and move 8 right.

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3.03 Practice questions Understanding

Explorer

Adventurer

1–5

Trailblazer

1–5

In which direction do we move on a number line if we are adding a positive integer?

In which direction do we move on a number line if we are subtracting a positive integer?

In which direction do we move on a number line if we are adding a negative integer?

In which direction do we move on a number line if we are subtracting a negative integer?

The expression 9 + (−4) can be thought of as “start with 9 and move 4 units to the left”. For each expression, name the starting number, the number of units to move and whether you move left or right on the number line. a

3 – 12

Fluency 6

−5 + 9

−2 + (−8)

6, 7a-fk, 8, 9a-h, 10a-h, 11

−4 − (−11)

6bdfh, 7d-j, 8, 9c-j, 10e-l, 11, 12

6bdfh, 7f-l, 8bdeh, 9e-l, 10i-p, 12

Use the plotted point on the number line to add these integers: a

5+9 −5

−15

−10

−5

−15

−10

−5

−15

−10

−5

−3 + 8

−6 + 3

−7 + 7

3+9+7 −5

−5

−7 + 7 + 4 −10

−8 + 6 + 9 −10

−5

−10 + 5 + 12 −15

−10

−5

3.03 Add and subtract integers mathspace.co

117

Evaluate: a

Ex 2

−9 + 0

−45 + 45

−39 + 97

−26 + 15

−13 + 8 + 2

−24 + 10 + 54

−88 + 28 + 92

Use the plotted point on the number line to subtract these integers:

118

−10

−5

−10

−5

−10

−5

7−9

3−3

−9 − 9 −20

−15

−10

−5

−15

−10

−5

−10

−5

−2 + 7 − 4

10 − 7 − 2

14 − 18 + 1 −10

−5

−25

−20

−15

−10

−5

11 – 11

−1 − 15 − 3

12 – 3

9 – 13

−13 − 13

−32 – 41

36 – 53

−9 − 5 – 8

−12 − 3 + 16

27 − 19 – 10

0 − 11 + 36

−7 − 8 – 15

14 − 49 + 35

Evaluate: a

14 + (–5)

−7 + (−8)

2 − (−3)

5 − (−7)

−12 − (−12)

17 + (−14)

29 − (−52)

−30 − (−32)

12 + 17 + (−15)

−12 + 16 + (−18)

17 − 27 + (−17)

−5 + (−2) + (−7)

m −9 − 2 − (−8)

−5 − (−2) − (−10)

−17 + 23 + (−20)

34 − 63 − (−34)

5 ÷ (20 − 15)

7 × (4 − 2) – 3

27 ÷ (5 − 2) + 4

Evaluate: a

8−5

Evaluate: a

−15 + 15

−21 + 0 + 21

−15 + 6

Ex 3, 4

−7 + 15

14 + 7

−3 + (7 × 5)

Determine whether the following statements are true or false: a

−5 + 8 ≥ 3 + (−2) + 5

−12 − 4 < −15 + 2 + 1

−9 + 6 > −3 + (−1) + 1

−4 + 3 ≤ −2 + (−2) + 2

Mathspace WA Curriculum – Year 7 mathspace.co

Reasoning 13

13, 14

14, 15

Evaluate: i

13, 16

6 + (−6)

(−8) + 8

Use the number line to explain your results from part (a).

What is the only number that is never written with a negative or positive sign? Why do you think this is?

If a positive integer is added to a negative integer, can the result ever be negative? Use an example to explain your reasoning.

If you add a negative integer to another negative integer, what is the sign of the result? Use examples to support your answer.

Problem-solving 17, 18, 20, 23, 24 17

19–23, 25–28

Valentina and Maria want to meet at the cinema to watch a movie. Valentina lives 3 km west of the cinema and Maria lives 5 km east of the cinema. a

NW W

Cinema

West Cinema East

West Maria’s house

Valentina’s house

West Cinema East Valentina’s house

In the number lines below, the distance between each tick is 1 km. Which of these number lines shows the location of Valentina and Maria in relation to the cinema?

Maria’s house

17–20, 23–26

E SW

SE S East

Valentina’s house

West

Maria’s house

Cinema Valentina’s house

East Maria’s house

Calculate how far Maria would have to travel to get to Valentina’s house.

The image below shows how the location of a miner travelling up and down a mine shaft relates to an integer on the number line: a

Which integer represents 3 m above the surface?

Which integer represents 4 m below the surface?

If Nadia is initially 2 m above the surface and descends 6 m in the elevator, which integer represents her new location?

If Matt is at a location represented by the integer −4 and ascends 3 m, describe his new location.

5 m above 4 m above 3 m above 2 m above 1 m above surface 1 m below 2 m below 3 m below 4 m below 5 m below 6 m below 7 m below 8 m below

3.03 Add and subtract integers mathspace.co

5 4 3 2 1 0 −1 −2 −3 −4 −5 −6 −7 −8

119

One evening in Moscow, the temperature fell to −3 °C. By midday the next day, the temperature was forecast to be 15 °C. Calculate the expected rise in temperature.

Julia dives off a platform that is 8 m above the water. She descends a total of 14 m before returning back up to the surface.

If Julia’s initial position on the diving platform is represented by the integer +8, calculate the integer that represents her lowest position throughout the dive.

Determine the greatest depth that Julia reaches during the dive.

A submarine is 150 m below sea level. It rises 40 m, then dives 75 m before stopping. Determine the distance of the submarine below sea level after it has stopped.

A hot air balloon starts at an altitude (height) of 500 metres then ascends (goes up) 100 metres, descends (goes down) 225 metres, then ascends 150 metres. What is the balloon’s ﬁnal altitude?

A bus departs the depot with 8 passengers on board. At the ﬁrst stop, 4 people got off and 6 people got on. At the second stop, a further 3 people got off and 7 more got on. Calculate the number of people that are on the bus when it leaves the second stop.

Liquid nitrogen can be used to make ice cream. The ice cream’s temperature starts at 30 °C, then rapidly cools to −180 °C after liquid nitrogen is added to it. Determine the integer that represents this change in temperature.

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Mathspace WA Curriculum – Year 7 mathspace.co

Three contestants are heading into the ﬁnal round of a game show. Eileen has 39 points and Roxanne has −3 points. If Oprah has 14 more points than Eileen, calculate how many points separate Roxanne and Oprah.

An integer has been assigned to each letter below. A word must be formed using these letters and each letter can only be used once. The table shows the assignment of the integers: Letter Integer

−3

−2

−1

The score of the word is the sum of the integers assigned to its letters. Determine the score of these words: a 27

NET

EAR

RAIN

CARES

Each bottle of orange juice is supposed to have a capacity of 250 mL. Ten bottles are inspected by the quality assurance department. The results of the inspection are shown in the table: Bottle Amount below or above the required capacity (mL) a b

−7

−11

−1

−5

Bottle 10

Determine the actual capacity of: i

Bottle 1

Bottle 2

iii

Bottle 6

Calculate the total capacity contained in the 10 bottles.

The motion of a robot to the left or right along a straight line is determined by whether the robot is facing left or right and whether it is stepping forwards or backwards. Use a number line where 1 unit represents 1 step and the positive direction is to the right to answer these questions: a

Facing Right Right Left Left

Stepping Forward Backward Forward Backward

Moving Right Left Left Right

Determine the integer that represents the final position: i

The robot starts at 0, facing right and takes 5 steps forward.

The robot starts at 0, facing left and takes 13 steps forward.

iii

The robot starts at 0, facing left and takes 11 steps backwards.

Suppose the robot performs these instructions in the given order: • Start at −5, facing right. • Take 7 steps forward. • Take 21 steps backward. • Turn to face left. • Take 22 steps backward.

Determine the integer representing the ﬁnal position.

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121

3 Chapter review 1

Determine whether each number is an integer or a non-integer: a

8.2

−7.35

−2.850

−4

1250

If you have two integers plotted on a number line, how do you know which integer is the smaller?

Identify the integer plotted on each number line: a

−5

−4 −3 −2 −1

0 0

−6

−3

−8

−4

d 2

Identify the integer that is three units to the right of 4.

Identify the integer that is four units to the left of 15.

Identify the integer that is three units to the left of −3.

Identify the integer that is five units to the right of −11.

Identify the integer that is 5 units to the right of −2.

Identify the integer that is 3 units to the left of 1.

For each number line: i

Arrange all the integers plotted in ascending order.

Identify the largest number.

−12

−8

−4

−25

−20

−15

122

−18, −6, −29

−9, −28, 12

0, −3, 7

38, −40, −5

Identify the inequality sign, < or >, that should appear between each pair of numbers to make each statement true: e

−5

Identify the smallest number in each set: a

−10

−10 ⬚ 10

−2 ⬚ −35

b f

Evaluate:

9 ⬚ −6

45 ⬚ −80

c g

−3 ⬚ −7

−17 ⬚ −15

d h

0 ⬚ −10

−50 ⬚ 25

18 + 9

−20 + 7

−32 + 14

−50 + 50

−35 + 85

−17 + 11 + 3

−30 + 15 + 45

−90 + 35 + 97

Mathspace WA Curriculum – Year 7 mathspace.co

Evaluate: a

18 − 5

−17 − 8

15 − 19

−22 − 36

7 − 13 + 29

−16 − 5 + 21

−9 − 7 − 12

20 − 45 + 25

Evaluate: a

8 − (−10)

−9 + (−12)

−14 − (−14)

18 + 20 + (−24)

−7 + (−4) + (−9)

−11 − 3 − (−10)

−18 + 22 + (−24)

40 − 72 − (−36)

8 ÷ (24 − 16)

5 × (7 − 3) − 4

36 ÷ (8 − 5) + 7

Evaluate: a

−5 + (9 × 6)

Consider the following statement: “climbing 18 metres upwards”. a

State the opposite of “climbing 18 metres upwards”.

Suppose “climbing 18 metres upwards” is represented by the number 18. Identify the integer that should represent its opposite.

Arrange the following altitudes in order from lowest to highest: −20 m, 25 m, −8 m, 5 m, −13 m

Arrange the following numbers in ascending order, separated by inequality notation. 8, −18, 22, −5, 13

Complete each number pattern: a c

3, 6, 9, ⬚, ⬚, ⬚

−5, −2, 1, ⬚, ⬚, ⬚

−14, −10, −6, ⬚, ⬚, ⬚

8, 5, 2, ⬚, ⬚, ⬚

One afternoon in Rome, the temperature fell to −5 °C. By late morning the next day, the temperature was forecast to be 12 °C. Calculate the expected rise in temperature.

A caver is 80 m below ground level. They climb 35 m, then descend 45 m before taking a break. Determine the depth of the caver below ground level after they have taken a break.

A factory starts the day with 15 boxes of materials available. At the first delivery, 7 boxes are used and 10 boxes are delivered to the factory. At the second delivery, a further 5 boxes are used and 12 more boxes are delivered. Calculate the number of boxes available at the factory after the second delivery.

Each container of nuts is supposed to contain a weight of 500 g. Nine containers are inspected and the results of the inspection are shown in the table: Container Amount below or above the required weight (g) a

−5

−12

+10

−2

−8

Container 10

Determine the actual weight of: i

Container 1

Container 2

iii

Container 6

Calculate the total weight contained in the 10 containers.

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Big ideas Fractions are essential for developing proportional thinking and reasoning.

Curriculum Outcomes • Add and subtract positive fractions with related and unrelated denominators using flexible and efficient strategies • Multiply and divide positive fractions using flexible and efficient strategies • Explore and represent equivalent fractions with related and unrelated denominators, visually and numerically • Draw and label, or use a given number line, to locate, order and compare with equality and inequality symbols, fractions, terminating decimals, percentages and integers • Explore and interpret multiplication and division of positive fractions, visually and numerically • Extend the use of associative, commutative and distributive laws, additive and multiplicative partitioning, inverse operations, order of operations, equality and inequality to validate a range of mental and written strategies involving the four operations on whole numbers, positive fractions and decimals, and addition and subtraction of integers • Analyse, represent and interpret problems in real-world situations involving whole numbers, positive fractions, decimals and percentages, addition and subtraction of integers, numbers in index form, linear equations with up to two operations, simple number patterns and/or transactional money statements

4 Fractions Chapter outline 4.01 Identify, order and compare fractions 4.02 Negative fractions 4.03 Equivalent and simplified fractions Investigation: Fraction flowers 4.04 Add and subtract fractions Investigation: Egyptian fractions 4.05 Fractions of a quantity 4.06 Multiply fractions 4.07 Divide fractions 4.08 Order of operations with fractions Chapter 4 review

126 134 140 150 158 164 170 177 182

Fractions are a baker’s best friend when it comes to scaling recipes and perfecting dessert sizes!

4.01 Identify, order and compare fractions After this lesson, you will be able to… • identify fractions and parts of a fraction. • differentiate between improper fractions and mixed numbers. • compare and order fractions.

Parts of fractions Part of this shape is shaded and part of it is not. Notice that the shape is made of ﬁve equal parts and only three of them have been shaded. We can call this amount .

The point marked on this number line is halfway between 0 and 1. That is, the point represents the number that is as far from 0 as it is from 1. Notice that the number line is split into two equal parts and the point lies at the end of the ﬁrst part. We can call this number .

These numbers are examples of fractions. The top part of a fraction is called the numerator. This tells us how many equal parts of the whole are present. The bottom part is called the denominator. This tells us the number of equal parts that are in one whole. The line in the middle is called the vinculum.

4 5

Numerator Vinculum Denominator

For example, consider

In this fraction, the numerator

is 4 and the denominator is 5. We can call this fraction 4 on 5, 4 over 5, 4 out of 5 or 4 ﬁfths.

Rational numbers can be made by dividing an integer by an integer. All fractions are a division of one integer by another integer, so all fractions are rational numbers.

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Example 1 What fraction of the hexagon is shaded?

Create a strategy Write the fraction as the number of shaded parts over the total number of parts.

Apply the idea The hexagon is divided into 6 parts. The number of shaded parts is 5. This means that 5 is the numerator and 6 is the denominator. So the fraction is

Example 2 What fraction is plotted on the number line? 0

Create a strategy

Apply the idea

The number of spaces between 0 and the point is the numerator. The number of spaces between 0 and 1 is the denominator.

There is 1 space between 0 and the point. So the numerator is 1. There are 5 spaces between 0 and 1. So the denominator is 5. The fraction is

Idea summary Fractions are rational numbers which can be written as , where a and b are integers. They are made from one or more equal parts of a whole number. A fraction is made up of: • A top number, called the numerator, indicates how many equal parts of the whole are present. • A bottom number, called the denominator, indicates how many equal parts are in one whole. • A line between the two numbers, called the vinculum.

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Improper fractions and mixed numbers Fractions like

make up less than one whole. We know this because the numerator is less than

the denominator. We call these proper fractions. What about a fraction like ? Notice that the numerator is greater than the denominator. This means that the fraction is greater than a whole. We call these improper fractions. Each rectangle in this image has been split into ﬁve equal parts, so 5 is the denominator. Eight parts have been shaded, so 8 is the numerator. Since the number of shaded parts is more than the number of parts in one whole, a complete rectangle and three more parts have been shaded. We can write this number as

, which we call one and three ﬁfths.

Numbers like this are called mixed numbers or mixed numerals. Mixed numbers and improper fractions can also be represented on a number line. For

each whole number on the number line is

split into 5 equal parts. 0

For the improper fraction, we would count 8 tick marks from 0 and place the point there. For the mixed number, we would count 3 tick marks from 1.

Example 3 What number is plotted on the number line? Give your answer as a mixed number. 3

Create a strategy For the fraction part, count the number of equal spaces between the two whole numbers, then count how many spaces after the previous whole number the point is located.

Apply the idea The point is between 3 and 4. It is greater than 3 but less than 4. So the whole number part is 3. There 10 equal spaces between 3 and 4, so each space represents right of 3, so the fraction is The number plotted on the number line is

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The point is 9 spaces to the

Example 4 Plot these on a number line and then order them in ascending order:

and

Create a strategy Create a number line that goes from 0 to 2, marked with eighths.

Apply the idea Create the number line and show it. is 1 whole and 5 more eighths. is 1 eighth less than 1 whole. is more than 1 whole; if we convert this to a mixed number, we can see that it is 1 whole and 3 more eighths.

Ordering in ascending order, this means from left to right, we have ,

and

Idea summary Fractions where the numerator is less than the denominator are called proper fractions. Fractions where the numerator is greater than the denominator are called improper fractions. Numbers which are made up of a whole number and a fraction are called mixed numbers or mixed numerals.

Compare fractions Which fraction is greater:

or ? The ﬁrst thing we can do is make a visual model for each fraction.

We can see that more of the

fraction bar has been shaded than the

fraction bar. Try creating

fraction bars for other fractions with a denominator of 8. Notice that the smaller the numerator, the smaller the fraction. This works for any two fractions with the same denominator.

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We could write this as

Which fraction is greater:

Here, more of

? Again, we can make a visual model for each fraction.

and has been shaded than

Try creating fraction bars for other fractions with a

numerator of 4. Notice that the smaller the denominator, the bigger the fraction. This works for any two fractions with the same numerator. We could write this as

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Example 5 Which inequality symbol makes this sentence true? A <

⬚ ?

Create a strategy

Apply the idea

Since the numerators are the same, the fraction with the greater denominator is the smaller fraction.

The numerators of the fraction are the same. The denominator of denominator of

is smaller than the

So we have

The answer is option B.

Idea summary Fraction bars can help us compare fractions. For any two fractions with the same denominator: the smaller the numerator, the smaller the fraction. For any two fractions with the same numerator: the greater the denominator, the smaller the fraction.

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4.01 Practice questions Understanding 1

Identify the numerator.

Identify the denominator.

Write each word as a fraction: a

One quarter

Three eighths

Sixteen hundredths

Two ninths

One and ﬁve tenths

Six and three ﬁfths

4, 5a-c, 6ace, 7a-c, 8a-c, 9a-d, 10a-d, 11a-c

4, 5c-e, 6ace, 7c-e, 8c-e, 9c-f, 10c-f, 11c-e, 12ab

4c, 5d-f, 6bdf, 7d-f, 8d-f, 9e-h, 10e-h, 11d-f, 12cd

What fraction of each shape is shaded? a

Ex 2

Trailblazer 1d, 2d, 3bdf

Write each fraction in words:

Fluency

Ex 1

a 3

Adventurer 1b, 2b, 3bdf

For each fraction: i

Explorer 1ac, 2ac, 3ace

What fraction is plotted on each number line? a

b 1

d 1

f 1

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Ex 3

For each group of shapes: i

Write the fraction that is shaded as a mixed number.

Write the fraction that is shaded as an improper fraction.

What number is plotted on each number line? Give your answer as a mixed number. a

c e

132

d 9

b 4

Ex 4

f 11

Plot these on a number line and then order them in ascending order: a

Express these mixed numbers as improper fractions: a

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 5

For each pair of fractions, determine which fraction is greater: a

Fill in the missing inequality symbol to make the statements true. a e

⬚

b f

Is each statement true or false? a

Reasoning

⬚

13, 15, 16

14–16

⬚

17, 18

Which fraction is greater, one third or one half?

Which fraction is smaller, one fourth or one ﬁfth?

Identify which fraction is greater:

If a number line from 0 to 2 is divided into eight equal parts, what is the fraction represented by the point that is halfway between 1 and 2? Explain your reasoning.

Which is greater,

? Explain your reasoning.

? Explain your reasoning and use a visual representation, such as a

number line or area model, to support your answer. 18

If a number line is divided into 15 equal parts and point A is located number line and point B is located

of the way along the

of the way along, what fraction represents the point that

is equally distant from A and B? Explain your reasoning using a number line.

Problem-solving 19ab, 20bd, 21 19

For each group of fractions, arrange them from largest to smallest: a

19cd, 20ac, 22

For each group of fractions, arrange them from smallest to largest: a

19bc, 20bc, 23

Petria has read

of her book and Suma has read

d of theirs. Who has read more of their book?

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Tristian is looking at data from 3 different surveys showing how many people agreed with the focus question. Which survey group shows the most support? a

20 people agreed from a total of 26 surveyed

21 people agreed from a total of 26 surveyed

21 people agreed from a total of 27 surveyed

Alice is preparing a series of 3 fruit smoothies for her friends. The smoothie recipes have different concentrations of fruits to liquids. The fraction of fruits in the smoothie are apple smoothie,

for the mango smoothie and

for the

for the strawberry smoothie. Order the

smoothies from least to the most fruit content in their recipes.

4.02 Negative fractions After this lesson, you will be able to… • compare and order positive and negative fractions on a number line. • interpret a fraction’s relative values.

Fractions on the number line On the number line shown, each tick is labelled with a multiple of the fraction the point furthest to the left is plotted at the fraction

We can see that

and the point furthest to the right is plotted

at the fraction

It is common to see number lines where only the integers are labelled, with ticks between each integer that represent a fraction of one whole.

−4

−3

−2

−1

A point has been plotted on this number line. Can you see which fraction the point lies on?

We can ﬁnd the denominator of the fraction using the number of ticks between each integer. To go from 0 to 1, we need to move 3 ticks, so each tick represents To ﬁnd the numerator of the fraction, we can count the ticks from 0 to the point, which gives 8. Since we moved in the negative direction, the point is plotted at the fraction Another way to identify the fraction is to see that the point is two thirds to the left of the integer −2. So, we can write it as the mixed number

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Example 1 Where is the point plotted on the number line? −2

−1

Create a strategy The number of spaces between the numbers is the denominator and the number of spaces between the point and 0 is the numerator.

Apply the idea There are 5 spaces between the numbers which means the denominator is 5. For the numerator, we can see that from 0, the number is 8 spaces to the left. This means it is 8 of the fifths away. But the point is to the left of 0 which means the fraction is negative. The number is

Reflect and check The number can also be written as

, since it is 3 spaces to the left of -1.

Idea summary The number of spaces between the integers is the denominator and the number of spaces between the point and 0 is the numerator of the plotted fraction on a number line.

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Fractions with different denominators If we are given two fractions that do not have the same denominator, seeing which is greater may not be obvious. For example, how can we compare

and

In the last lesson, we used fraction bars to compare fractions, but we can also use number lines. First, we can plot the point that represents the fraction −1

by dividing each whole into 6 equal

parts. Then, we move 5 ticks to the left of 0. The point on this number line is at Next, we can plot the point that represents the fraction

−1

by dividing each whole into 3 equal

parts. Then, we move 2 ticks to the left of 0. The point on this number line is at

If we compare the number lines, we notice that the integers are aligned and that than

Because

is further left

is closer to 0, we know it is the greater fraction. Another thing to notice is that these denominators are related. Since 6 is a multiple of 3, we can use the same number line to compare these fractions. The longer ticks are marked at thirds, but the shorter ticks are marked at sixths.

Using the number line, we can see that

−2

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is the same as

and

is the same as

Now that we have common denominators, we can plot the points on the same number line as shown.

Example 2 Which fraction is greatest? Use number lines to assist you. A

Create a strategy Find the least common multiple of the denominators to get the common denominator for the three fractions. Then plot the fractions on the same number line to compare.

Apply the idea Notice that the least common multiple of 12, 3 and 6 is 12. This means we need to make each fraction have a common denominator of 12. The fraction

has already a denominator of 12.

The fraction

is the same as

The fraction

is the same as

Plot each fraction on the number line like this:

Since

is in the furthest right,

is the greatest number.

The answer is option C.

Idea summary To compare fractions with different denominators, we can use two different number lines. We can also ﬁnd common denominators to be able to compare on the same number line.

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4.02 Practice questions Explorer

Understanding

1–3

Trailblazer

2, 3

Identify which of these fractions are negative and explain your reasoning:

What fraction does each tick represent on this number line? −3

−2

−1

Compare and contrast the representation of

Fluency Ex 1

Adventurer

4–9, 10bd, 11bc

and

on a number line.

4–9, 10bc, 11ab

4, 6–9, 10ac, 11ad

Identify the fraction plotted on each number line: a

b −2

−1

−4

−3

−2

−1

−3

−2

−1

For each pair of fractions: i a

138

Plot the fractions on the number line. and

and

−1

−2

Identify which one is closer to zero. 0

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Mathspace WA Curriculum – Year 7 mathspace.co

Identify the greatest fraction plotted on each number line: a −2

−1

b −1

Identify the smallest fraction plotted on each number line: a −7

−6

−5

−4

−3

−2

−1

b −4

Ex 2

−3

−2

−1

Arrange these fractions in ascending order:

Arrange these fractions in descending order:

Identify the greatest fraction in each set of fractions. Use number lines to assist you. a

Identify the smallest fraction in each set of fractions. Use number lines to assist you. b

Reasoning

12, 13

12, 14

Explain why the fractions that represent “hiking north” would be on opposite sides of 0.

Which of these negative fractions is closest to zero? Explain your reasoning.

Arrange these negative fractions in ascending order. Explain your reasoning.

Problem-solving 15 15

km due south” and “hiking

A toy car is moving along a straight track. Its initial position is at and it moves to a new position at

km due

15, 16

from the starting position

from the start. How far did the toy car move?

Express the distance as a positive fraction. 16

A submarine is located 600 metres below sea level. It descends another is the submarine’s new depth in negative metres?

of its depth. What

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4.03 Equivalent and simplified fractions After this lesson, you will be able to… • find equivalent fractions. • simplify fractions. • compare fractions.

Equivalent fractions Sometimes different fractions can represent the same amount. Consider the shaded area of the hexagon below. We can see that 3 of the 6 parts have been shaded so the area represents

of the whole shape.

We can also see that one half, , of the shape has been shaded. Since the same area is shaded for both fractions must be equal.

, these two

These are called equivalent fractions, since they have different numerators and different denominators but are still equal.

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We can see equivalent fractions in action by representing them visually and seeing how we can create them. Consider this grid: This grid is divided into 6 equal parts as shown by the solid grid lines. 1 out of 6 parts is shaded, so we can represent this area as

of the grid.

If we separate each of the 6 parts into 3 more equal parts, as shown by the dashed lines, we can create an equivalent fraction.

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Notice there are now 18 grid squares and 3 of them are shaded. This means that we can represent the shaded area with the fraction

This shows that

are equivalent fractions, which we can express using an equal sign:

By separating each of the original parts into 3 more equal parts, we are multiplying the numerator (the shaded part) and denominator (the entire grid) by 3:

Example 1 Rewrite

with a denominator of 40.

Create a strategy We need to find what number to multiply 8 by to reach 40. 8 × [ ] = 40. Then multiply the numerator and denominator by this value.

Apply the idea 40 ÷ 8 = 5. Multiply the numerator and denominator by 5

Evaluate

Idea summary When two fractions represent the same amount of a whole, they are equivalent fractions. We can create an equivalent fraction by multiplying or dividing the numerator and denominator by the same number.

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Simplify fractions We can also create equivalent fractions by decreasing the denominator. To decrease the denominator, we can remove common factors in both the numerator and denominator. Since the numbers become smaller, we call this simplifying the fraction. Previously, we saw that

From here, we can remove the common factor of 3:

This shows:

The fraction

has no common factors between the numerator and denominator (other than 1), so it

is fully simpliﬁed.

Example 2 In the grid shown,

parts have been shaded. Use the grid to simplify

Create a strategy Find the fraction of columns that is shaded.

Apply the idea The grid can be divided into 8 identical columns as pictured shown.

5 out of 8 columns are shaded. So,

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Reﬂect and check We also could have simpliﬁed the fraction by writing the numerator and denominator as a multiple of a common factor and then removing the common factor. The common factor of 20 and 32 is 4 so we can write both as multiples of 4: Write both numbers as multiples of 4

Remove the common factor of 4

Example 3 Select the two fractions which are equivalent to

Create a strategy To find equivalent fractions, find common denominators between

and each option.

Common denominators will share the same factors.

Apply the idea a)

and Look for ways to simplify both fractions. ;

simplifies to

;

simplifies to

This shows that b)

is equivalent to

and We just saw that

, these are not equivalent fractions.

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143

and ;

simplifies to

We already know that

, these are not equivalent fractions. d)

and We have already determined that

and

are equivalent. This is the second answer.

Reflect and check For option C, we could also have seen that with

because

and then comparing

we would see that they are not equivalent.

Idea summary When we find an equivalent fraction by removing common factors from the numerator and denominator, we are simplifying the fraction. When the fraction has no common factors between the numerator and denominator (other than 1), it is a fully simplified fraction. Consequently, two fractions are equivalent when they can be simplified to the same fraction.

Compare fractions To compare fractions with different denominators, we can ﬁnd equivalent fractions with the same denominator and compare the equivalent fractions. For example, which is greater:

We can try comparing these fractions visually:

5 parts out of 12 Because the size of

3 parts out of 8

is different from the size of , it is difficult to compare the fractions. To make

it easier, we can find equivalent fractions with the same denominator. But what denominator should we try to get?

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When comparing fractions, we want to make their new denominators equal to the lowest common multiple of their current denominators. Since 24 is the lowest common multiple of 12 and 8, we want to find the equivalent fractions of

and that have a denominator of 24.

We can do this by multiplying the numerator and denominator of denominator of

Since

by 2 and the numerator and

by 3.

is greater than

, we can see that

is greater than .

We could also add some gridlines to the fraction grids so that they both have 24 equal parts in order to compare them.

10 parts out of 24

9 parts out of 24

We can see that after adding the gridlines, the grid for now has

shaded parts. This tells us that

now has

shaded parts and the grid for

is greater than .

Example 4 Arrange these fractions from greatest to smallest:

Create a strategy Find a common denominator of the fractions.

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Apply the idea A common multiple of 3, 7 and 5 is 3 × 5 × 7 = 105. Multiply the numerator and denominator by 35

Evaluate

Multiply the numerator and denominator by 15

Evaluate

Multiply the numerator and denominator by 21

Evaluate

Arranging these from greatest to smallest, we have:

So we have this order for the original fractions:

Idea summary To compare fractions with different denominators: • Find the lowest common denominator • Find the equivalent fraction with this denominator • Compare the numerators of the equivalent fractions

4.03 Practice questions Understanding

Explorer 1–4

Adventurer 1, 3, 4

Trailblazer 1, 4, 5

Which of these fractions are fully simplified?

How do you compare two fractions with the same denominator, such as

What is the first step when comparing two fractions with different denominators?

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and ?

Complete the solution to simplify the fraction

Which of these statements are correct? A

A fraction is in its simplest form if the numerator and denominator have common factors other than 1.

A fraction is in its simplest form if the numerator and denominator have no common multiple.

A fraction is in its simplest form if the numerator and denominator have no common factors other than 1.

A fraction is in its simplest form if the numerator is 1.

Fluency Ex 1

Ex 2

6a-f, 7a-d, 8ab, 9, 10a, 11aceg, 12ac, 13a, 14a

6c-h, 7c-f, 8bc, 9, 10b, 11cefg, 12ac, 13a, 14a

6e-l, 7e-h, 8cd, 10bc, 11bdeh, 12bd, 13b, 14b

Complete each equivalent fraction statement: a

Use the diagrams provided to simplify these fractions: a

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Ex 3

Determine whether each fraction is equivalent to a

Simplify: a

For each pair of fractions, identify which fraction is greater: b

a Ex 4

Identify all the equivalent fractions from each list: a

For each group of fractions: i

Find the lowest common denominator of the fractions.

Rewrite all the fractions with this lowest common denominator.

iii

List the fractions in descending order.

For each group of fractions: i

Find the lowest common denominator of the fractions.

Rewrite the fractions with this lowest common denominator.

iii

List the fractions in ascending order.

Reasoning

15, 16

15, 17

17, 18

Are

Determine an equivalent fraction to

Which of these pairs of fractions are equivalent? Explain your reasoning.

If the numerator and denominator of a fraction are both prime numbers, is the fraction in fully simplified form? Explain your answer.

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and

equivalent fractions? Explain your reasoning. that has a denominator of 12. Explain your reasoning.

Problem-solving 19, 20 19

19, 20

20, 21

Four out of a class of 24 students have blue as their favourite colour. One sixth said that green was their favourite. If

said that they preferred red, create a circle graph showing how

many students like each of the colours and determine if any students did not answer. 20

If a pizza has been cut into 8 slices and 3 slices have been eaten, what fraction of the pizza remains? What is an equivalent fraction for the remaining fraction with a denominator of 16?

A group of students is collecting data on their favourite colours. They found that 5 out of 20 students like blue. There are 28 students in the class next door which has the same proportion (or fraction) of students who like blue. Write an equivalent fraction for students who like blue for the class next door.

Did you know?

You can explore fractions in a fun and artistic way through something called “Fraction Flowers”! Each flower is made up of colorful petals, where each petal represents a different fraction of the whole. For example, one flower might have 2 large petals that each represent , while another flower might have 4 smaller petals, each representing . By arranging the petals into full circles, you can see how different fractions come together to make a whole. Fraction flowers are a great hands-on way to understand how fractions work together and can be used to compare different fractions visually. They’re not only educational but also creative, making maths feel like art!

Investigation: Fraction flowers

Investigate online

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4.04 Add and subtract fractions After this lesson, you will be able to… • add and subtract fractions with the same denominators. • add and subtract fractions with different denominators. • add and subtract mixed numbers.

Add and subtract fractions with the same denominator Suppose we want to ﬁnd

. Here we have a circle with 2 sevenths shaded and a circle with 3 sevenths shaded. Notice that the parts of each circle are the same size. Since the parts are the same size, we can place one circle on top of the other. Now, we can see 5 sevenths of the circle are shaded. So, we can conclude that

When the denominators are the same, we are adding quantities of the same size. This means we can add the number of shaded pieces, without needing to change the number of parts in the whole. Mathematically, we are adding the numerators but keeping the same denominator. Suppose we want to ﬁnd

. Using the same circles as above, we can take 2 sevenths away from 3 sevenths. The part that remains is 1 seventh of the circle. So, we can conclude that

When the denominators are the same, we are subtracting quantities of the same size. Again, the number of shaded pieces is changing, but the number of parts in a whole stays the same. Mathematically, we are subtracting the numerators but keeping the same denominator.

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Example 1 Evaluate

and simplify your answer.

Create a strategy Add the numerators over the same denominator and simplify.

Apply the idea Add the numerators

Rewrite with a common factor of 2

Remove the common factor

Idea summary When two fractions have the same denominator, we can add or subtract them by adding or subtracting the numerators, while keeping the denominator the same.

Add and subtract fractions with different denominators If the denominators of two fractions are different, then we are not adding quantities of the same size.

Interactive exploration

Discover this concept in action online

mathspace.co

We can use area models to change fractions to equivalent fractions, so the denominators are the same. Then, we can add the equivalent fractions. Consider

. These two fractions look like this. If we tried to add the pieces together, how could we write the denominator? We cannot place one circle on top of the other because the number of parts in a whole are different.

Before we can add these two fractions, we need to rewrite them with the same denominator. To do this, we create equivalent fractions. The denominators here are 9 and 4, so the lowest common multiple of the denominators is 4 × 9 = 36. This is sometimes called the lowest common denominator. 4.04 Add and subtract fractions mathspace.co

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Now, we can rewrite the fractions with a denominator of 36.

The fractions can now be divided into the same number of parts. After shading the correct amount of pieces, the fractions look like this.

Since the sizes of the parts are the same, we can add the fractions together.

This shows that

When the denominators are different, we create equivalent fractions with the same denominator, then add or subtract the numerators.

Example 2 Evaluate

Create a strategy Find the lowest common multiple of the two denominators.

Apply the idea The lowest common multiple of 4 and 8 is 8. Since the denominator of

is already 8, we only need to ﬁnd the equivalent fraction of . Multiply the numerator and denominator by 2

Evaluate Substitute the new fraction

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Subtract the numerators

Mathspace WA Curriculum – Year 7 mathspace.co

Idea summary When two fractions have different denominators, we ﬁrst rewrite the fractions with the same denominator. Then, we can add or subtract the numerators and keep the denominator the same.

Add and subtract mixed numbers Mixed numbers have a whole number part and a fractional part. There are two methods for adding or subtracting mixed numbers: 1. Add or subtract the whole parts, then add or subtract the fractional parts. • If the fractional parts do not have the same denominator, we need to rewrite them with the same denominator before adding or subtracting. 2. Convert the mixed numbers into improper fractions, then add or subtract the improper fractions. • If the denominators are different, we need to rewrite them with the same denominator before adding or subtracting.

Example 3 Evaluate

Create a strategy Add the whole number parts and the fraction parts separately.

Apply the idea Group the whole numbers and the fractions

Add the whole numbers and the fractions

Write as a mixed number

Example 4 Fill in the boxes to complete the working out:

a Rewrite as improper fractions.

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Create a strategy Multiply the whole number by the denominator, then add the numerator.

Apply the idea Multiply the whole number by the denominator

Evaluate the multiplication

Evaluate the addition

b Rewrite with a common denominator.

Create a strategy Find the least common multiple (LCM) of the denominators and multiply each numerator and denominator by the quotient obtained by dividing the LCM by their respective denominators.

Apply the idea The LCM of 3 and 4 is 12. So we need to multiply both parts of

by 12 ÷ 3 = 4 and multiply both parts of Multiply

and

by 12 ÷ 4 = 3.

Evaluate the multiplication

c Evaluate the difference.

Create a strategy Evaluate the answer from part (b).

Apply the idea Subtract the numerators

Idea summary To add or subtract mixed numbers, we can write them as improper fractions, create equivalent fractions with the same denominator, then add or subtract them. Another method of adding or subtracting mixed numbers is to rewrite the fractional parts to have the same denominator, then add or subtract the whole parts and add or subtract the fractional parts.

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4.04 Practice questions Understanding 1

Explorer 1–4

Adventurer

Trailblazer

1–4

3, 4

Which is correct?

Describe how to add fractions with common denominators.

What is the first step when adding or subtracting fractions with different denominators?

One method to add mixed numbers such as and then add the fraction parts.

is to first add the whole number parts

Describe another method to add mixed numbers.

Fluency 5

5ac, 6b-df, 7, 8abef, 9–13, 14a-f

5d, 6e-h, 8cdgh, 10–11, 13, 14e-h, 15bd

Find the lowest common denominator of each set of fractions: a

Ex 1

5bc, 6aceh, 7ac, 8abef, 9–13, 14e-h, 15ac

Evaluate and simplify:

Complete each statement: Rewrite with a common denominator

Evaluate the sum

Rewrite with a common denominator

Evaluate the sum

Rewrite as a mixed number

Rewrite with a common denominator

Evaluate the difference

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Rewrite with a common denominator

Evaluate the difference Ex 2

Rewrite as a mixed number

Evaluate and simplify: a

Fill in the boxes to complete the working out: Group the whole number parts and fraction parts

Add the whole number parts and fraction parts

Evaluate the sum

Fill in the boxes to complete each equation: b

For

Find the sum of the whole number part of each mixed number.

Find the sum of the fraction part of each mixed number.

Simplify the expression.

Fill in the boxes to complete the working out: Rewrite as improper fractions

Rewrite with a common denominator

Evaluate the difference

Ex 3, 4

156

For

Rewrite the mixed numbers as improper fractions.

Using part (a), rewrite the improper fractions with the same denominator.

Simplify the expression.

Evaluate and simplify: a

Mathspace WA Curriculum – Year 7 mathspace.co

Evaluate and simplify: a

Reasoning 16

For

16, 17

16, 18

16, 18, 19

Calculate the answer by first converting both mixed numbers to improper fractions.

Calculate the answer by first subtracting the whole numbers and then subtracting the fraction parts.

Which method is less likely to cause errors? Explain your answer.

Teddy and Archie are sharing a pizza while watching movies. Teddy takes

and Archie

takes . What fraction of the pizza is left? Explain the steps to solve the problem. 18

A recipe calls for

cup of sugar, but Wei only has

cup of sugar. How much more sugar

does Wei need to complete the recipe? Is your answer reasonable? Explain. 19

A rectangular cake measures 9 cm by 8 cm. You cut a piece that measures 4 cm by 3 cm. What fraction of the cake is left after cutting this piece? Explain your reasoning using a diagram.

Problem-solving 20–23

20, 23, 24

20, 22, 24, 25

Jack felt dehydrated. He drank 1 third of a litre at ﬁrst, then drank another 1 ninth of a litre. How many litres did he drink in total?

Jane eats

of a cake and Joshua eats

of the cake. What fraction of the cake did they eat

altogether? 22

During a blizzard, it snowed

. When the sun came out the next day,

of snow

melted. How much snow was left on the ground? 23

A market stall is open for

hours in the morning and

hours in the evening. How many

hours is the stall open altogether? 24

Georgio buys 3 cups of sugar to use in his baking. He uses cup in his pastries and accidentally spilled

of a cup in his cake,

of a cup. How much sugar does he have left?

Missy mailed 4 letters that have a total weight of 3 grams. One letter weighed another weighed

of a

of a gram and a third weighed

of a gram,

of a gram. How much did the fourth

letter weigh?

Investigation: Egyptian fractions

Investigate online

mathspace.co

4.04 Add and subtract fractions mathspace.co

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4.05 Fractions of a quantity After this lesson, you will be able to… • use multiplication to find groups of fractions and fractions of a quantity. • describe parts of a quantity and learn to use multiplication to find fractions of a quantity. • simplify fractions by removing common factors in the numerator and denominator.

Groups of fractions Suppose we want to ﬁnd 6 groups of . We can imagine having 6 whole circles, then shading of each circle. If we count each of the shaded parts, we will ﬁnd there are 12 thirds, which can be combined to make whole circles.

Similar to whole numbers, we can think about 6 groups of

as repeated addition or multiplication.

Recall that when we add fractions of the same size, we only add the numerator because the number of parts in a whole does not change. This means we only multiply the numerator by 6:

Example 1 Find 3 groups of

Create a strategy The phrase “groups of” means multiplication.

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Apply the idea Multiply the numerator by the whole number

Evaluate

Idea summary Finding a whole number of groups of a fraction is the same as repeatedly adding the fraction to itself. Since repeated addition is multiplication, we can multiply the whole number by the numerator and keep the denominator the same.

Fractions of a quantity Fractions describe parts of one whole, but they can also describe parts of a quantity. For example, to ﬁnd

of 36, we are ﬁnding a fraction of a quantity. Let’s start by drawing 36 squares. To ﬁnd 5 twelfths of the squares, we need to split them into 12 equal groups.

Looking at the 12 groups, each contains 3 squares. We need to know what

of 36 is, which is the number of

squares in 5 of the 12 groups. Since there are 3 squares in each part and we are interested in 5 parts, we can multiply 3 × 5 to get 15 squares. This means We can also work this out arithmetically. We can write

of 36 is 15. of 36 using multiplication:

In the example above, we started by dividing 36 into 12 equal groups. We can represent this mathematically by removing the common factor:

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This gave us the number of squares in each group. Since we were interested in ﬁve of the groups, we multiplied the number of squares in each group by 5: 5 × 3 = 15 Just like before, we could have multiplied the whole number by the numerator and simpliﬁed the fraction: Multiply the numerator by the whole number

Evaluate the multiplication

Rewrite using a common factor of 12

Remove the common factor

Whether you remove the common factor first or last, both methods lead to the same answer.

Example 2 Evaluate

Create a strategy Multiply the numerator by the whole number.

Apply the idea Multiply the numerator by the whole number

Evaluate

Simplify

Example 3 Evaluate

of 150 L.

Create a strategy of 150 L is the same as convert to an improper fraction.

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, when multiplying by a mixed number we first need to

Apply the idea Convert

to an improper fraction.

Reflect and check As a quick check for reasonableness we can check what 2 groups of 150 L and 3 groups of 150 L is.

is between 2 and 3. So our answer should be: • greater than 2 groups of 150: 2 × 150 = 300 L • smaller than 3 groups of 150: 3 × 150 = 450 L

Example 4 Find

of 5 weeks in days.

Create a strategy The word “of” means multiplication. Use the fact that 1 week = 7 days.

Apply the idea Convert 5 weeks to days: 5 weeks = 5 × 7 days = 35 days

Multiply 5 by 7 Evaluate

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So we have: Substitute the number of days

Multiply the numerator by the whole number

Evaluate

Simplify

of 5 weeks is 25 days.

Idea summary Finding the fraction of a quantity is the same as multiplying a fraction by a whole number. It is often easier to remove common factors in the numerator and denominator before evaluating the multiplication.

4.05 Practice questions Understanding

Ex 2

Adventurer 1, 3

Explain the process of ﬁnding

Write two thirds of 15 as a multiplication.

equivalent to

Fluency Ex 1

Explorer 1, 3

162

Trailblazer 2, 3

Explain why or why not. of 50 in two ways: using a diagram and multiplying fractions.

4a-d, 5a-f, 6bd-f, 7a-df-j

4abgh, 5g-l, 6bdfh, 7f-n

4d-g, 5g-l, 6acgh, 7ej-p

Evaluate: a

28 groups of

40 groups of

4 groups of

18 groups of

84 groups of

6 groups of

4 groups of

64 groups of

Evaluate and simplify: a

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 3

Ex 4

Evaluate: a

Find these quantities in the given units: a

of $1 in cents

of 6 minutes in seconds

of 4 weeks in days

of 6 L in millilitres

of 3 kg in grams

of 3 cm in millimetres

of 5 m in centimetres

of 36 g in milligrams

of 3 hours in minutes

of 12 km in metres

of $69 in cents

of 1200 mL in litres

of 8 hours in minutes

of 45 days in hours

of 27 weeks in days

of 300 g in kilograms

Reasoning 8

8–10

Justin and Georgia are handing out ﬂiers to advertise their new concert. Justin has handed out 120 of his 400 ﬂiers while Georgia has handed out

9–11

of her 600 ﬂiers.

What fraction of his ﬂiers has Justin handed out?

How many ﬂiers has Georgia handed out?

Who has handed out a higher fraction of their ﬂiers?

How is the process for ﬁnding the fraction of a quantity in parts (a) and (b) different?

A school has 480 students and

of the students are in the drama club. How many students

are in the drama club? Explain your reasoning. 10

A jar contains 500 marbles, of which are blue and the rest are red. How many red marbles are in the jar? Explain your reasoning.

A fruit bowl contains 12 pieces of fruit. of pears is

are apples and there are 4 bananas. The number

of the number of bananas. The rest are oranges. How many oranges are in the

bowl? Yulia’s work is shown. Identify the row with the error then correct the step to get the correct answer. 1

Number of apples

Number of pears

Number of oranges 4.05 Fractions of a quantity mathspace.co

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Problem-solving 12, 14, 16

13, 15–17

13, 14, 17–19

Xavier received $10 pocket money and spent did he spend?

Sinead read

In a class of 27 students, in the class?

Charlie won $40 000 in the lottery. He gave did Charlie keep?

A Year 6 class planned to raise $400 for a local charity by Christmas. If they’ve already raised

of it on mobile games. How much money

of a 450 page book. How many pages has she read? of the students have dark hair. How many students have dark hair

of his winnings to his sister. How much money

of the money, how much more do they still need to raise?

A biased coin lands on tails

of the time.

Why is the coin unfair?

If the coin was ﬂipped 40 times, how many times would it be expected to land on tails?

If a fair coin was ﬂipped 40 times, how many times would you expect it to land on tails? Why?

The Fireﬂies scored 77 points in their basketball game. Fe scored scored

of the points and Jamie

of the points. Sarah was the only other scorer. How many points did Sarah score?

In the morning, a dam contains 1710 L of water. Throughout the day, lost through evaporation,

through cattle drinking from it and

of the total water is

is used for watering crops.

How many litres of water are lost throughout the day?

At the end of the day, how many litres of water remain in the dam?

4.06 Multiply fractions After this lesson, you will be able to… • visualise multiplication of fractions using diagrams and models. • multiply mixed numbers with fractions.

Multiply fractions We’ve seen how to ﬁnd fractions of quantities, but can we ﬁnd fractions of parts?

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In the previous lesson, we found each of 6 circles.

Now suppose we want to ﬁnd

of 6 by shading

of . We can apply the same ideas. We will start with a diagram of . Our goal is to shade

in each of the already shaded ﬁfths.

To do this, we need to split each of these ﬁfths into thirds. Notice that the circle is now divided into ﬁfteenths (that is, 3 × 5) and twelve parts have been shaded (3 × 4). We know from equivalent fractions that this tells us

Then, we can shade 2 thirds of each of the originally shaded ﬁfths. Now, we see that

If you look closely, you can see that the numerators of the two original fractions have been multiplied to get the numerator of the answer and the denominators of the two original fractions have been multiplied to get the denominator of the answer.

In general, whenever we want to multiply two fractions, we multiply the numerators and the denominators separately. Sometimes we might have to simplify the resulting fraction afterwards.

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Example 1 Evaluate

Create a strategy Multiply numerators and denominators separately.

Apply the idea Multiply numerators and denominators

Evaluate

Simplify

Example 2 Evaluate

Create a strategy Simplify the numerator and denominator with the same multiple then evaluate.

Apply the idea Remove common factor of 5

Remove common factor of 3

Remove common factor of 4

Evaluate the multiplication

Idea summary To multiply fractions, multiply the numerators and the denominators separately.

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Multiply with mixed numbers Recall that a mixed number is a number consisting of an integer and a fraction. We can write mixed numbers as a whole number added to a fraction. For example,

To multiply a mixed number by a fraction, we could use the distributive law of multiplication. For example:

From here, we would need to evaluate the multiplication and add the resulting fractions. We could do this since these are skills we have been practising, but there is a faster and simpler way. Remember that mixed numbers can also be written as improper fractions. If we do this conversion ﬁrst, we can do the multiplication in fewer steps: Convert the mixed number to an improper fraction

Evaluate the multiplication

Notice that we can use the same steps for multiplying fractions that we previously saw, with one added step. That is, we can convert mixed numbers to improper fractions ﬁrst.

Example 3 Evaluate

Create a strategy Rewrite the mixed numbers as improper fractions.

Apply the idea Rewrite mixed numbers as improper fractions

Remove common factor of 3

Remove common factor of 7

Evaluate

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Example 4 Danielle takes minutes to drive from her home to the local shopping centre. She spends this time waiting at traﬃc lights.

Find the number of minutes she spends waiting.

Create a strategy

Apply the idea

Multiply the given numbers.

Multiply

Rewrite

as improper fraction

Evaluate

Idea summary To multiply mixed numbers by a fraction or by another mixed number, convert the mixed number to an improper fraction ﬁrst. Then, multiply the numerators and denominators separately.

4.06 Practice questions Understanding 1

Explorer

Adventurer 1b, 2, 3

1–3

Describe how to multiply

and

using a rectangular area model.

Describe how to multiply

and

using multiplication of numerators and denominators.

For these sets of multiplications of fractions:

Why is any fraction multiplied by a fraction in the form 3

168

Trailblazer 1b, 2, 3

equal to itself?

Identify if the statements are true or false for positive fractions: a

Multiplying a fraction by a fraction that is greater than 1 increases the original fraction.

Multiplying a fraction by a fraction that is greater than 1 decreases of the original fraction.

The product of two proper fractions is always less than each of the original fractions.

The product of two proper fractions is always more than each of the original fractions.

Mathspace WA Curriculum – Year 7 mathspace.co

Fluency 4

Ex 1

Ex 2

Ex 3

4aceg, 5aceg, 6aceg, 7acdg, 8–13

4acefg, 5aceg, 6aceg, 7bceg, 10–13

4bdefh, 5bdefh, 6bdfh, 7e-h, 10–14

Evaluate: a

Evaluate:

Simplify:

of a cake was left after a party. Hank ate

of what was left. What fraction of the whole

cake did Hank eat? 9

Georgia was following a recipe which requires

of a cup of ﬂour. However, she only

wants to make half of the quantity that the recipe is for. What fraction of a cup of ﬂour should she use? 10

In a milkshake shop, a medium milkshake needs

of milk. A small milkshake needs

the amount of a medium milkshake. How much milk does the small milkshake need? 11

A farmer has

of an acre of land. If he wants to plant crops on

of the land, how much of

the land will he use for planting? How much land will be left unused? Ex 4

Stephen is years old. He has spent he spent living in Sydney?

In Wings Station, one batch of chicken wings requires is making

of his life living in Sydney. How many years has

batches this Sunday, how much ﬂour will they need?

Natalia is starting her jewellery business. She bought weighs

cups of ﬂour. If the Wings Station

bags of beads. If a single bag

grams, what is the total combined weight of the beads Natalia bought?

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Reasoning

15, 16

A recipe calls for

16, 17

cup of sugar, but you only want to make one-third of the recipe.

Describe the steps you could take to ﬁnd how much sugar you need. Cynthia explains that if each juice box contains

will contain

cup of juice, then a pack of 6 juice boxes

times that amount of juice. She multiplies

cup by

to ﬁnd the total amount

of juice in the pack. Amount of juice in the pack Is Cynthia correct? Why or why not? A rectangular garden measures

metres by

metres. What is the area of the garden in

square metres? Will 10 square metres of turf cover the garden? Explain your reasoning.

Problem-solving 18, 19 18

Lianne had have left?

Opie ordered

18–20

L of orange juice to sell. She sold

18–20

of it. How much orange juice does she

of apples to sell. He found that

of the apples were not fit for sale.

What weight of apples could he sell? 20

A recipe for a cake requires

cup of sugar to make 1 cake.

How much sugar is needed to make 5 cakes?

Josie ate

If 10 cups of sugar costs $2.50, how much will it cost to buy enough sugar to make 5 cakes?

of these cakes, how many cups of sugar is this?

4.07 Divide fractions After this lesson, you will be able to… • divide fractions by whole numbers. • divide fractions by other fractions. • divide fractions by mixed numbers.

Divide fractions by whole numbers When we divide whole numbers, we split the whole number into equally sized parts and ﬁnd the size of one part.

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For example, consider 6 ÷ 3.

This is similar to asking, “If we divided 6 into 3 groups of the same size, what would be the size of each group?” Since there are 3 equal groups of 2 in 6 we can say that 6 ÷ 3 = 2.

Recall that fractions represent division, so 6 ÷ 3 can be written as . In the previous lessons, we saw

This means that dividing by 3 is the same as multiplying by .

Now consider

. This is similar to asking, “If we divided 1 2

into 3

groups of the same size, what would be the size of each group?” Looking at the whole, we see it is divided into 6 pieces. So, the size of each group is .

Just like we saw above, we can say This isn’t a coincidence. Dividing by any number is the same as multiplying by the reciprocal of that number. The reciprocal of a whole number is the fraction of one over that whole number. If we think of the whole number as a fraction over one, that is, numerator and the denominator.

, the reciprocal is found by swapping the

Example 1 Evaluate

. Give your answer as a fraction.

Create a strategy Rewrite the whole number as a fraction. Then multiply the ﬁrst fraction by the reciprocal of the second fraction.

Apply the idea Rewrite the whole number as fraction

Multiply by the reciprocal of

Multiply the numerators and denominators

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Idea summary Dividing by any number is the same as multiplying by the reciprocal of that number. The reciprocal of a whole number can be found by expressing the number as a fraction with 1 as the denominator, then swapping the numerator and denominator.

Divide by fractions If dividing by a whole number is like multiplying by a fraction, what is dividing by a fraction like? Consider

. We cannot ask, “What is 6 divided into

groups of the same size” because that

does not make sense. Instead, we can say, “How many groups of 0

0 3

2 3

4 3

6 3

8 3

10 3

12 14 3 3

16 3

18 3

are in 6?”

To answer this question, we can begin by dividing the 6 wholes into thirds. Then, we can make groups of 2 thirds. If we count, we can see there are 9 equal groups of

in 6. This shows

Again, rather than dividing by a fraction, we can multiply by the reciprocal of the fraction:

The same reasoning applies when we divide fractions by fractions. Consider the problem This is like asking, “How many groups of

are in

First, we can draw a diagram of .

Then, we can split it into groups of . We can see that there are 4 equal groups of So,

in .

Once again, we can check to see that dividing by a fraction is the same as multiplying by the reciprocal of the second fraction:

Note that we must only take the reciprocal of the second fraction. The ﬁrst fraction always stays the same. 172

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Example 2 Evaluate

. Give your answer as a fully simpliﬁed fraction.

Create a strategy Divide fractions by multiplying the ﬁrst fraction by the reciprocal of the second fraction.

Apply the idea Multiply by the reciprocal

Multiply numerators and denominators

Evaluate

Simplify

Idea summary To divide by a fraction, multiply the ﬁrst number by the reciprocal of the fraction.

Divide by mixed numbers Similar to multiplying with mixed numbers, it is possible to use models to show division with mixed numbers, but it can be complicated. When dividing with mixed numbers, it is much faster to convert them to improper fractions ﬁrst, then divide the fractions like we did before.

Example 3 Evaluate

, writing your answer in its simplest form.

Create a strategy Rewrite the mixed numbers as improper fractions, then divide the fractions.

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173

Apply the idea Rewrite as improper fractions

Multiply by the reciprocal of

Simplify

Multiply numerators and denominators

Example 4 A

metre long roll of fabric is to be cut into sections of equal length

. How many pieces of

fabric will there be?

Create a strategy Rewrite the mixed numbers as improper fractions then perform division.

Apply the idea Rewrite as improper fractions

Multiply by the reciprocal of

Remove the common factor

Simplify by evaluating the division

There will be 3 pieces of fabric.

Idea summary When dividing with mixed numbers, convert them to improper fractions ﬁrst. Then, multiply the ﬁrst number by the reciprocal of the second number.

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4.07 Practice questions Understanding

Explorer 1–3, 5

What is the reciprocal of: b

Write the problem

Can you divide a fraction by itself? If so, explain what the result will be.

Fill in the boxes to complete each equation: a

Fluency

Ex 3

1–5

Describe the relationship between dividing fractions and multiplying fractions.

Ex 2

1–5

Trailblazer

Ex 1

Adventurer

as a multiplication.

6aceg, 7a-f, 8abfh, 9–12, 14

6aceg, 7a-f, 8c-f, 11–15

6bdfh, 7g-l, 8d-g, 11–15

Evaluate and simplify: a

Evaluate and simplify:

A family drinks family?

A glass holds of a litre of lemonade. How many glasses of lemonade can be ﬁlled with 6 litres of lemonade?

of a carton of milk per day. How many days will 6 cartons of milk last the

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of a tart is shared equally between 5 people, what fraction of the tart would each person

receive? 12

Yang eats

of a packet of biscuits each day. If he buys 12 packets, how many days will the

packets last? 13

Christine has

of a metre of ribbon. She uses

ths of the ribbon to wrap a present.

How many presents can she wrap with her ribbon? Ex 4

Sheldon is planning to sell banana bread at the upcoming school fair. If a loaf requires cups of sugar, how many banana bread loaves can Sheldon make if he has

cups of sugar

on hand? 15

Jean is making bows for her gift wrapping business. A bow requires ribbon. If she has

Reasoning 16 17

centimetres of

centimetres of ribbon, how many bows can Jean make? 16, 17

If you need to divide

16, 17

17, 18

by , which fraction do you take the reciprocal of and why?

You have a pizza that is divided into 8 equal slices and you want to give

of a slice to each

of your friends. Describe the steps you could do to ﬁnd how many friends you can share your pizza with. 18

Sam and Ruby are given the following problem: If it takes book?

of an hour to read

of a book, how many minutes will it take to read the entire

Sam says it will take 67.5 minutes and Ruby says it will take 53.3 minutes. Explain who is correct and why.

Problem-solving 19 19

19, 20

A water tank has a capacity of 1500 litres. If the tank is currently ﬁlled to

of its total capacity?

Benjamin wants to engrave his name on a metal plate. There is

cm of space on the plate.

There are three sizes of letters to choose from: a

cm wide,

cm wide and

cm wide.

Find the maximum number of letters that Benjamin can engrave on the plate using the following sizes: i

176

of its capacity,

how much more water needs to be added to the tank to reach

cm wide

iii

Which letter sizes could he use for his name?

Mathspace WA Curriculum – Year 7 mathspace.co

cm wide

4.08 Order of operations with fractions After this lesson, you will be able to… • simplify expressions with multiple operations involving fractions. • apply the order of operations to fractions. • solve expressions involving fractions.

Order of operations with fractions Recall the order of operations is a set of rules that help us know how to correctly simplify an expression with multiple operations. The order of operations with fractions is the same as the order of operations with whole numbers: • Evaluate operations inside grouping symbols such as brackets. • Evaluate multiplication or division going from left to right. • Evaluate addition or subtraction going from left to right. There are a few things to keep in mind when using the order of operations with fractions: First, it is best to write integers as fractions with a denominator of 1. Second, it is best to convert mixed numbers into improper fractions before doing any calculations. Lastly, it is important to remember that when adding and subtracting fractions we need to have common denominators, but this is not the case for multiplying and dividing.

Example 1 Evaluate and simplify

Create a strategy Evaluate the multiplication, then evaluate the addition.

Apply the idea Evaluate the multiplication

Multiply the numerators and the denominators

Evaluate the addition

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Example 2 Evaluate and simplify

Create a strategy Follow the order of operations. Evaluate the operation in the brackets, then evaluate the subtraction (going from left to right).

Apply the idea Rewrite both fractions with same denominator

Evaluate all multiplications

Evaluate the subtraction inside the brackets

Evaluate the subtraction

Example 3 Evaluate and simplify

Create a strategy Evaluate the operation in the brackets, then the division.

Apply the idea Rewrite the mixed number as improper fraction

Rewrite the fraction to have same denominator

Evaluate the multiplication

Evaluate the subtraction

Rewrite the whole number as fraction

Multiply by the reciprocal

Multiply the numerators and denominators

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Idea summary The order of operations with fractions is the same as the order of operations with whole numbers: • Evaluate operations inside grouping symbols such as brackets. • Evaluate multiplication or division going from left to right. • Evaluate addition or subtraction going from left to right.

4.08 Practice questions Understanding 1

Explorer 1–2, 3a-e

Adventurer 2, 3a-e

Trailblazer 3cdf-h

Arrange these steps into the correct order of operations: • Evaluate addition or subtraction from left to right • Evaluate expressions inside the brackets • Evaluate multiplication or division from left to right

How do the brackets in

For each expression, what operation should be performed first? a

Fluency Ex 1

Ex 2

change the order of operations from the expression

4a-e, 5a-ek, 6a-d, 7adeh, 8ab, 9a-c, 10ab

4c-g, 5d-i, 6c-f, 7ad-f, 8ac, 9cef, 10b-d

4d-h, 5f-l, 6e-h, 7bcfg, 8bd, 9d-f, 10d-f

Evaluate and simplify: a

Evaluate and simplify:

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Ex 3

Evaluate and simplify: a

Evaluate and simplify:

Evaluate and simplify: a

Evaluate and simplify:

Identify whether each statement is true or false: a

Reasoning

11, 12

12, 13

Simplify

Riley and Reese are given the following problem:

12, 13

. Explain what operation you did ﬁrst and why.

There are 12 students in a scout group.

of them will be going on a trail walk. 20 students

from another scout group are meeting them at the trailhead. In total, how many students are going on the walk? Riley uses the expression

to solve the problem and Reese uses the expression

to solve the problem. Will Riley and Reese get the same result? Explain your reasoning.

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To ﬁnd the area of the composite shape shown, Ruby uses the expression

1 2

Jade uses the expression 3

1 4 2

Will Ruby and Jade get the same result for the total area of the composite shape? Explain your reasoning.

Problem-solving 14

14, 15

A survey shows that a quarter of the people in a town prefer chocolate ice cream, a third prefer vanilla and the remaining 300 people prefer strawberry. How many people were surveyed?

A school choir needs to divide If they spend

hours of rehearsal time between three different songs.

of their time on the ﬁrst song and

of their remaining time on the second

song, how much time do they have left to work on the third song?

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4 Chapter review 1

For each shape identify the fraction that is shaded and the fraction that is unshaded: a

For each group of shapes: i

Write the fraction that is shaded as a mixed number.

Write the fraction that is shaded as an improper fraction.

What fraction is plotted on each number line? a

b 1

d 7

Plot each fraction on the number line: a

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Is each statement true or false? b

a 6

Plot the fractions on the number line. and

Determine which one is closest to zero.

and

Complete each equivalent fraction statement: a

Determine if each faction is equivalent to a

What is the smallest fraction in each set of fractions? a

For each pair of fractions: i

Select the equivalent fractions from these lists: a

c 11

For each group of fractions: i

Find the lowest common denominator of the fractions.

Rewrite all the fractions with this lowest common denominator.

iii

List the fractions in descending order.

Evaluate and simplify: b

a 13

For

Find the sum of the whole number part of each mixed number.

Find the sum of the fraction part of each mixed number.

Simplify the expression.

After a heavy rainstorm, out the next day,

of water accumulated on the ground. When the sun came of water evaporated. How much water was left on the ground?

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Mishy mailed 4 letters that have a total weight of another weighed

of a gram and a third weighed

grams. One letter weighed

of a gram,

of a gram. How much did the fourth

letter weigh? 16

Evaluate: a

At Sweet Delights, one batch of chocolate chip cookies requires Delights is making

cups of sugar. If Sweet

batches this weekend, how much sugar will they need?

Evaluate and simplify: a

A large container of lemonade can fill

cups. If each serving at a party requires

cups of

lemonade, how many servings can be provided with the large container? 20

A recipe for a chocolate cake requires

cups of cocoa powder. If Sarah has

cups of

cocoa powder available, how many chocolate cakes can she make? 21

A car’s fuel tank can hold

litres of petrol. If the car consumes

litres of petrol for every

100 kilometres, how many kilometres can the car travel with a full tank of petrol? 22

To make a batch of homemade pizza dough, John needs

cups of flour. If he has

of flour, how many batches of pizza dough can John make? 23

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Evaluate and simplify: a

Mathspace WA Curriculum – Year 7 mathspace.co

cups

“The art of doing mathematics consists in finding that special case which contains all the germs of general methods.” Mary Cartwright

Big ideas Decimals represent quantities that are a combination of whole numbers and fractional parts. They provide a way to represent and manipulate numbers between whole numbers, offering a more precise and flexible system for dealing with realworld quantities, measurements and calculations.

Curriculum Outcomes • Convert between fractions, decimals and percentages using flexible and efficient strategies • Multiply and divide positive decimals using flexible and efficient strategies • Use appropriate rounding, estimation strategies and context to check reasonableness of solutions • Explore and interpret multiplication and division of positive decimals, visually and numerically • Use place value understanding to explore rounding decimals to a specified number of decimal places • Extend the use of associative, commutative and distributive laws, additive and multiplicative partitioning, inverse operations, order of operations, equality and inequality to validate a range of mental and written strategies involving the four operations on whole numbers, positive fractions and decimals, and addition and subtraction of integers • Identify the features of transactional statements and verify transactions. Explain reasons for checking and keeping financial records

5 Decimals Chapter outline 5.01 5.02 5.03 5.04 5.05 5.06

Decimals and place value Add and subtract decimals Convert between fractions and decimals Multiply decimals Divide decimals Round decimals Investigation: Account statements Chapter 5 review

188 194 200 209 216 221 229

Decimals are essential for measuring the growth rate of coral reefs, which can be as slow as 0.8 cm per year!

5.01 Decimals and place value After this lesson, you will be able to… • recognise the place value of digits in decimal numbers. • compare decimals using place value tables. • multiply and divide decimals by powers of ten.

Place value of decimals Decimals are used to represent values that fall between whole numbers. A decimal point is used to separate the whole number from the decimal or fraction part.

ne O

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When we work with decimals, we need to extend our understanding of place value to include tenths, hundredths, thousandths and beyond. Each digit after the decimal point represents a fraction of the whole number.

1000

100

1 10

1 100

1 1000

In 312.45, the digit 3 represents 3 hundreds, and the digit 4 represents 4 tenths. As with whole number place values, the farther to the left a column is, the larger its place value. Hundreds 3

Tens 1

Ones 2

. .

Tenths 4

Hundredths 5

Example 1 What is the numeric value represented by the digit 7 for 48.07?

Create a strategy Write the number in the appropriate place value column.

Apply the idea Tens 4

Ones 8

. .

Tenths 0

Hundredths 7

The digit 7 is in the hundredths place. So, it represents 7 hundredths or 0.07.

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Thousandths

Idea summary When we look at a number, we should pay attention to the value of each digit. Each digit to the right of the decimal point represents a fraction of the whole number. Hundreds

Tens

Ones

. .

Tenths

Hundredths

Thousandths

Compare decimals We compare decimal numbers using place values like we do with whole numbers. When we compare integers we know that 123 is greater than 19. How about comparing numbers between 0 and 1? Which is larger, 0.123 or 0.19? One of the easiest ways to compare numbers with decimals is to write both numbers with the same number of digits after the decimal point. We need to represent these numbers in a place value table to better understand their values. If we put both numbers in the same place value table, it becomes obvious when we compare columns from left to right: Ones 0 0

. . .

Tenths 1 1

Hundredths 2 9

Thousandths 3 0

The ﬁrst two columns have the same value, so we look at the next column to the right. 9 hundredths is larger than 2 hundredths, so 0.19 is larger than 0.123.

Example 2 Choose the larger decimal: A 2.305

2.503

Create a strategy Write the numbers in a place value table and compare the numbers.

Apply the idea Ones

Tenths

Hundredths

Thousandths

As we can see in the table, both numbers have the same value in the ones column so we move to the tenths column. In the tenths column, we can see that 5 is larger than 3. So 2.503 is larger than 2.305. So the correct answer is option B: 2.503.

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Example 3 Order these numbers in ascending order: 0.476, 0.2959, 0.5385.

Create a strategy Use a place value table and compare the numbers. Ascending means smallest to largest.

Apply the idea Write the decimals in a place value table and use zeros as placeholders. Ones 0 0 0

. . . .

Tenths 4 2 5

Hundredths 7 9 3

Thousandths 6 5 8

Ten thousandths 0 9 5

In the tenths column, we can see that 2 < 4 < 5. This means that 0.2959 < 0.476 < 0.5385. So the number in ascending order is: 0.2959, 0.476, 0.5385

Idea summary A place value table is useful when comparing numbers. To compare numbers involving decimals, put both numbers in the same place value table and compare columns from left to right.

Multiply and divide by powers of ten Multiplying and dividing by powers of 10 is made easier because we use a number system that is based on 10. When we multiply by 10, we just move all the digits one place to the left and when we divide by 10, we move all the digits one place to the right. Let us consider 1.823 × 10 Hundreds

Tens

Ones

Tenths

Hundredths

Thousandths

We can see that 1.823 × 10 = 18.23. Multiplying by 10 increases the place of each digit by one place.

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This time, let’s look at 73.6 ÷ 100. Hundreds

Tens

Ones

Tenths

Hundredths

Thousandths

We have to put in a zero if there is no digit in the ones place.

Dividing by 100 decreases the place of each digit by two places, thus 73.6 ÷ 100 = 0.736. You may have to add or remove zeros and you may have to add or remove the decimal point after you have ﬁnished moving the values. This depends on which columns the values end up in.

Example 4 What is 0.65 × 10? Write your answer in decimal form.

Create a strategy

Apply the idea

Use a place value table to visualise the value of each digit in the number. When we multiply a number by 10, each digit becomes 10 times larger and will move to the left one column.

Here is 0.65 in a place value table. Ones 0

. .

Tenths 6

Hundredths 5

Multiplying it by 10 means moving each digit one column to the left. Ones 6

. .

Tenths 5

Hundredths

0.65 × 10 = 6.5

Example 5 What is 8.4 ÷ 100? Write your answer in decimal form.

Create a strategy

Apply the idea

Use a place value table to visualise the value of each digit in the number. When we divide a number by 100, each digit becomes 100 times smaller and will move to the right two columns.

Here is 8.4 in a place value table. Ones 8

. .

Tenths Hundredths Thousandths 4

Dividing it by 100 means moving each digit two columns to the right. Use 0 as a placeholder. Ones 0

. .

Tenths Hundredths Thousandths 0 8 4 8.4 ÷ 100 = 0.084

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Idea summary Multiplying by a power of 10 increases the place value of each digit by the same number of zeros as in the power of 10. Dividing by a power of 10 decreases the place value of each digit by the same number of zeros as in the power of 10.

5.01 Practice questions Understanding

How many tenths are in 1 one?

How many hundredths are in:

1 one

1 tenth

1 thousandth

4–6, 7ace, 8a, 9b, 10af, 11a-f

4–6, 7ace, 8b, 9a, 10af, 11a-f

4–6, 7b-e, 8b, 9a, 10g-l, 11g-l

84.07

0.487

47.008

40.708

0.71 or 0.7

0.39 or 0.4

1.506 or 1.605

4.9999 or 4.8

0.5874, hundredths place

45.236, tenths place

71.029, thousandths place

2365.012, ones place

Insert the symbol > or < to make the expression true.

2.305 ⬚ 2.309

0.39 ⬚ 0.389

20.9999 ⬚ 20.9998

0.005 ⬚ 0.01

3.68, 3.86, 3.068, 3.681, 3.1

0.483 ⬚ 0.4

Write these numbers in ascending order. a

192

iii

1 hundredth

Identify the digit in the specified value for each number:

Ex 3

Choose the larger decimal between each number: a

1 tenth

Write down the place value represented by the digit 7 in these numbers: a

Fluency

Ex 2

Trailblazer 1–3

Match the written and numeric form of each number. i

Adventurer 1–3

Ex 1

Explorer 1–3

0.24, 0.204, 2.4, 0.024, 0.0024

Mathspace WA Curriculum – Year 7 mathspace.co

Write these numbers in descending order. a

Ex 4

Ex 5

0.54, 0.504, 0.5, 0.054, 0.0054

7.28, 7.82, 7.078, 7.168, 7.8

Evaluate, writing your answer in decimal form. a

0.56 × 10

5.2 × 10

0.35 × 100

6.74 × 100

0.6489 × 1000

5.761 × 1000

0.3784 × 10

0.003 × 10

9.3 × 100

59.43 × 100

0.04 × 1000

0.008 × 1000

6.75 ÷ 10

4.8 ÷ 100

Evaluate, writing your answer in decimal form. a

34 ÷ 10

0.832 ÷ 10

953.5 ÷ 100

7583.5 ÷ 100

0.57 ÷ 10

3.9 ÷ 10

79.66 ÷ 100

423 ÷ 1000

423.3 ÷ 1000

2.2 ÷ 1000

Reasoning

12, 13, 15

13–15

Consider the decimal numbers 3.45 and 3.045. Explain the process you would follow to determine which number is smaller. Discuss the role of place value in your comparison process.

Consider the product 0.973 × 100. a

Indicate which direction the digits must be moved to calculate the answer.

Identify the number of places the digits must be moved. Explain your answer.

Evaluate the product.

Explain why the process of multiplying and dividing decimal numbers by powers of 10 is different from multiplying and dividing by other numbers. Use examples to support your explanation.

Consider these two numbers: 0.672 33 and 0.8. Explain why the misconception “longer is larger” does not hold true in this case.

Problem-solving 16 16

16–18

A weather forecaster records the maximum temperature in Melbourne on ﬁve consecutive weekdays starting on Monday. She records the following temperatures: 17.5 °C, 23.8 °C, 19.6 °C, 21.2 °C, 15.9 °C

Which day was the hottest?

On which two days was it colder than the previous day?

Four students in a Year 7 class complete a quiz on decimals. Nicky scored 84.5, Daniel scored 91.3, Bernice scored 78.9 and Dylan scored 86.2. a

Order the students in descending order of score.

Cheska completes the quiz the following day and scores 91.25. What is Cheska’s rank in the class?

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In a number-guessing game, players need to identify a secret four-digit decimal number. They are given four hints: • The digit in the tenths place is twice the digit in the thousandths place. • The digit in the hundredths place is half the digit in the tenths place. • The digit in the ones place is the sum of the digits in the tenths and hundredths places. • The digit in the thousandths place is 3. Find the four-digit decimal number.

5.02 Add and subtract decimals After this lesson, you will be able to… • add and subtract decimals using the vertical algorithm, including regrouping and carrying over and using zeros as placeholders when necessary. • apply your skills in adding and subtracting decimals to solve real-world problems involving decimals.

Add decimal numbers To add decimals, we can use the same vertical algorithm or vertical addition approach that we use with whole numbers. We just need to consider the value of the digits and where our decimal point needs to be. When we add numbers with decimals, like 8.2 + 3.5, we use a place value table. This time, we need to include place values to the right of the decimal point. We also need to put the decimal point in its own column between the ones and the tenths. This helps us keep track of the value of each digit when we add them up. Tens Ones 8 + 3 1 1

. Tenths . 2 . 5 . 7

In this case, when we added the values in the ones column, we carried over into the tens column, giving us a ﬁnal answer of 11.7. Be sure to line up the numbers, so that the decimal points are in line, as this makes sure that the numbers with the same place value are in the same place column.

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Example 1 Evaluate 644.969 + 229.713

Create a strategy

Apply the idea 6 14 14 + 2 2 9 8 7 4

We can use vertical addition, being sure to line up the place value columns and decimal points.

. 9 16 9 . 7 1 3 . 6 8 2

Example 2 Evaluate 12.1 + 43.02 + 31.004

Create a strategy

Apply the idea

We can use vertical addition, line up the decimal points and place values and add 0 where place value holders are needed.

1 4 + 3 8

2 3 1 6

. 1 0 0 . 0 2 0 . 0 0 4 . 1 2 4

Idea summary Adding decimals uses the same process as we use for whole numbers, but we need to consider the value of the digits and where our decimal point needs to be.

Subtract decimal numbers When we subtract numbers with decimals, we again follow the same process as working with whole numbers and use place values. We are going to work through a subtraction where we need to regroup. For example, we could exchange 1 tenth for 10 hundredths. Let’s try 3.12 − 0.04. Use the vertical algorithm and align the decimal points. 3 . 1 2 – 0 . 0 4 In the hundredths column, 2 is less than 4, so we need to take away 1 from the tenths and carry it to the ones, which gives us 12. 3 . 0 12 – 0 . 0 4

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So now we have 12 − 4 = 8: 3 . 0 12 – 0 . 0 4 8 Subtract the tenths, 0 − 0 = 0: 3 . 0 12 – 0 . 0 4 0 8 Bring down the decimal point and subtract the ones, 3 − 0 = 3: 3 . 0 12 – 0 . 0 4 3 . 0 8 3.12 − 0.04 = 3.08 Sometimes when we subtract numbers with decimals, we need to use zeros as placeholders. For example, let’s subtract 87 − 5.45. –

8 7 . 0 0 5 . 4 5 8 1 . 5 5

Example 3 Evaluate 59 − 2.57, leaving the answer in decimal form.

Create a strategy Use the vertical algorithm, align the decimal points and use zero placeholders where needed.

Apply the idea We can add a decimal point and two zeros as placeholders in writing 59 so we can easily subtract the tenths and hundredths by 2.57. –

5 9 . 0 0 2 . 5 7

Subtract the values down each column, regrouping where necessary. 5 8 . 9 10 – 2 . 5 7 5 6 . 4 3

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Idea summary Subtracting numbers with decimals is like subtracting whole numbers, but we need to be careful with the decimal point. We need to line up the decimal points and subtract the numbers in each column. If there are any empty spaces, we can fill them with zeros to make it easier.

5.02 Practice questions Understanding

Explorer

Adventurer

1, 2

How does the process of adding decimals differ from adding whole numbers?

What is wrong with the way this question has been written down?

Fluency 3

3aceg, 4a-df, 5a-c, 6, 8, 10a-dg, 11a-e, 12, 14

2 . 6 . 9 9

3aceg, 4c-f, 5c-e, 6, 8, 10c-g, 11c-g, 12, 14, 16

3bdfh, 4e-h, 5c-f, 7, 9, 10d-h, 11d-h, 13, 15, 16

Evaluate: a

0.3 + 0.1

3.8 + 4.6

8.7 + 4.8

0.7 + 0.01

0.2 + 0.36

8.8 + 5.11

0.35 + 0.21

1.8 + 3.9

8.174 + 6.074

Evaluate: a

Ex 2

1, 2

– 1

Ex 1

Trailblazer

1, 2

2.414 + 1.586

9.99 + 50.39

84.03 + 64.08

91.71 + 76.5

207.791 + 120.201

221.219 + 716.225

714.969 + 129.713

Evaluate: a

0.10 + 0.25 + 0.30

0.16 + 0.8 + 0.76

0.91 + 0.829 + 0.117

8.23 + 6.52 + 9.16

4.027 + 1.39 + 1.306

15.1 + 53.04 + 21.002

Increase 13.245 by 4.62.

Find the amount that is $8.95 greater than $24.88.

Find the total of the numbers 0.325, 6.05, 12.066 and 0.004.

Calculate the sum of the numbers 1.55, 3.05, 105 and 0.05.

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Ex 3

Evaluate: a

9.9 − 8

5.9 − 1.7

5.4 − 2.3

10.6 − 4.9

8.5 − 7.2

0.83 − 0.61

0.75 − 0.51

3.1 − 0.6

Evaluate: a

8.79 − 4.5

58 − 1.57

9.59 − 1.7

6.19 − 5.31

64.51 − 53.47

81.59 − 45.71

0.587 − 0.231

73.893 − 64.764

Decrease 108.24 by 14.72.

Find the number that is 6.75 less than 27.18.

Find the difference between $91.33 and $22.44.

Reduce 132.035 by 56.122.

Harry buys an item from the school canteen for $3.20. If he pays for it with a ﬁve dollar note, how much change will he get back?

Reasoning 17

17ab, 19

17bc, 19

17cd, 18, 19

A magic square is a square where every row, column and diagonal adds up to the same ‘magic’ number. Complete these magic squares, given the magic number: a

Magic number is 3.3

1.2 1.1

Magic number is 9.0 2.4 5.4

0.7

3.0

0.6 c

3.6

Magic number is 6.0

Magic number is 10.2

2.4

0.6

3.9

3.6 3.2

1.8 3.3

1.2

2.1

4.8 1.5 2.7

Take three numbers between 1 and 10, each with two decimal places and add them together to get 22.54.

A customer gives the cashier $100 to pay for a purchase of $64.95. How much change should the cashier give the customer? Solenn incorrectly calculated that the change was $36.15. Identify in which step Solenn made the ﬁrst error and then correct her working. Step 1 1 0 0 . 0 0 6 4 . 9 5

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Step 2

9 10 . 10 10 – 6 4 . 9 5

Step 3 9 10 . 10 10 – 6 4

Step 4

9 10 . 10 10 – 6 4 . 9 5 . 1 5

Step 6

9 10 . 10 10 – 6 4 . 9 5 3 6 . 1 5

. 9 5 5

Step 5 9 10 . 10 10 – 6 4 . 9 5 6 . 1 5

Problem-solving 20, 21, 24

21, 22, 24

22, 23

Amy is buying some supplies for the new school year. She buys an exercise book for $9.55, a pack of pens for $6.12 and a pencil case for $6.28. How much does she spend in total?

Quentin recorded the following times in the 100 m running race over a whole season.

Race

Time (seconds)

1 2 3 4 5

10.59 11.03 10.45 10.38 10.63

How much quicker was his fastest time than his slowest time?

In a local food drive competition, three groups of volunteers collected non-perishable food items over a three-week period. Using the table, which group collected the most food items at the end of the three weeks?

Group 1

Group 2

Group 3

Week 1

5.28 kg

5.55 kg

2.208 kg

Week 2

4.15 kg

3.315 kg

10.19 kg

Week 3

4.325 kg

4.905 kg

1.63 kg

Three items weighing 1.22 kg, 4.66 kg and 2.77 kg are to be posted but are found to be over the 8.11 kg limit. By how much does the total weight exceed the limit?

A company is assessing the efficiency of four different manufacturing processes, each producing the same product. The processes have these defect rates: • Process A has a 0.023 defect rate • Process B has a 0.032 defect rate • Process C has a 0.0225 defect rate and • Process D has a 0.025 defect rate a

Rank the processes from the lowest to the highest defect rate.

Determine which two processes have the smallest difference in their defect rates.

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5.03 Convert between fractions and decimals After this lesson, you will be able to… • use place value to convert fractions with denominators that are powers of 10 to decimals. • use either the equivalent fraction method or division to convert fractions to decimals. • convert decimals to fractions using the place value method and simplifying the fraction.

Fractions to decimals using place value

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We already know that in the place value table, we have columns for tenths, hundredths, thousandths and so on. When we change a fraction to a decimal, it’s easiest if the denominator is a power of 10, so it matches our place value table.

1000

100

1 10

1 100

1 1000

To convert the fraction to a decimal, we need to write the digits in the correct column of the place value table, starting from the left. For example, if the denominator is 100, we write the digits in the hundredths column. If the denominator is 1000, we write the digits in the thousandths column and so on. Then we read the decimal number from left to right.

Example 1 Write the fraction

as a decimal.

Create a strategy Use a place value table.

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Apply the idea means 47 hundredths. So we can put this in a place value table: Ones 0

. .

Tenths 4

Hundredths 7

Idea summary If the denominator of a fraction is a power of 10, write the digits in a place value table to convert it into a decimal.

Convert fractions to decimals If we have a fraction that does not have a power of 10 in the denominator, there are two strategies we can use to convert the fraction to a decimal. One strategy to ﬁnd an equivalent fraction is when the denominator is easily converted to a power of 10. To do this, we can follow these steps: 1. Find a suitable number to multiply or divide the denominator by to make it a power of 10. 2. Multiply or divide both the numerator and denominator by this number. 3. Using a place value table, write out the number in decimal form. For example, if we want to convert equal to 100, a power of 10.

to decimal form, multiplying by 25 will make the denominator

The second strategy is useful when we cannot easily change the denominator to a power of 10. We can use division to divide the numerator by the denominator to ﬁnd the decimal. To convert

to a decimal, we can divide 5 by 16:

We can see that

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Example 2 Write the fraction

as a decimal.

Create a strategy Multiply the numerator and denominator of the fraction by a number that will make the denominator a power of 10.

Apply the idea Multiply both numerator and denominator by 4

Evaluate

Write as a decimal

Example 3 Convert

to a decimal rounded to the thousandths place.

Create a strategy Use short division, adding extra zeros as required to the numerator.

Apply the idea Set up the division including zero placeholders, up to one more place value than we are rounding to. In this case, 4 zeros. 11 does not go into 7 so we put a 0 above the 7 and copy the decimal point to the top. 11 goes into 70 six times since 11 × 6 = 66, so we put a 6 above the first zero and carry the remaining 4 to the next zero. Now, 11 goes into 40 three times, so we put a 3 above the second zero and carry the remaining 7 to the next zero. Continuing this pattern two more times, 11 goes into 70 six times, so we put a 6 above the third zero and carry the remaining 4 to the next zero. Then, 11 goes into 40 three times, so we put a 3 above the fourth zero. So if we round to the thousandths place, we have

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Example 4 Convert

to a decimal.

Create a strategy A mixed number has a whole number component. First, convert the fractional part part, 4.

to a decimal. After that, add this decimal to the whole number

Apply the idea Set up the division including zero placeholders, up to one more place value than we are rounding to. In this case. 3 zeros. 8 does not go into 3, so we put a 0 above the 3 and copy the decimal point to the top. 8 goes into 30 three times since 8 × 3 = 24. so we put a 3 above the first zero and carry the remaining 6 to the next zero. Now, 8 goes into 60 seven times, so we put a 7 above the second zero and carry the remaining 4 to the next zero. Continuing this pattern one more time, 8 goes into 60 five times, so we put a 5 above the third zero. So we have

Replace

with the decimal equivalent 0.375

Evaluate

Idea summary A decimal representation for a fraction can be found by either • •

Identifying an equivalent fraction that uses a power of 10 denominator Using a fraction as a division where we can divide the numerator by the denominator

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Convert decimals to fractions We can use place value to help us convert between decimals and fractions. With this method, we can transform any decimal into a fraction with a denominator of 10, 100, 1000 and more. Using place value, we know that 0.1 is equal to

because the 1 is in the tenths place. We can use

the same place value reasoning to see that 0.17 is equal to

We often prefer to write fractions in the simplest form. To simplify the fraction, we need to divide both the numerator and denominator by the highest common factor. For example, if we write 0.36 in a place value table like this: Ones 0

. .

Tenths 3

Hundredths 6

We can identify the place value of the decimal is hundredths and 0.36 can be written as . We can then simplify this by dividing both the top and bottom of the fraction by the highest common factor of 4:

Example 5 Convert these decimals to fractions: a Write the decimal 0.29 as a fraction.

Create a strategy Identify the place value of the decimal. In this case, it is hundredths.

Apply the idea Since there are two decimal places, the denominator of the fraction is 100. Write as a fraction

b Write the decimal 0.535 as a simplified fraction.

Create a strategy Identify the place value of the decimal. In this case, it is thousandths.

Apply the idea Since there are three decimal places, the denominator of the fraction is 1000. Write as a fraction

Divide both numerator and denominator by 5

Evaluate

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c Convert 4.535 to a fraction.

Create a strategy 4.535 is a mixed number that has 4 wholes and a 0.535 decimal part. We can use the 0.535 decimal to fraction conversion from part (b) and add 4 wholes to the fraction.

Apply the idea from part (b) So 4.535 is

Reflect and check For any decimal that has a whole number component, first convert the decimal part and then add the whole number component to the mixed number.

Idea summary To convert decimals to fractions, identify the place value of the decimal to rewrite the decimal as a fraction and simplify if needed.

5.03 Practice questions Understanding 1

Explorer

Adventurer

1–4

Trailblazer

2–4

means 47 hundredths. Put this in the place value table: Ones

. .

Tenths

Hundredths

What is the first step for converting fractions, such as

What is the ﬁrst step to convert a decimal to a fraction?

What is the denominator of the fraction for a decimal with 2 decimal places?

Fluency

5-7, 8a-hm, 9a-c, 10ab, 11a-eij, 12a-c, 13a-d, 14a-d, 15a-j, 16, 17

and

, into decimals?

5–7, 8e-n, 9ac, 10bc, 11d-j, 12c-e, 13c-f, 14c-e, 15e-m, 18

5–7, 8g-p, 9cd, 10cd, 11f-l, 12f-h, 13e-h, 14c-f, 15gk-p, 19

Write the number “ﬁfty-three hundredths” as a decimal.

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Ex 1

Consider this diagram: a

What fraction of the row of blocks is shaded?

Write this fraction as a decimal.

What fraction of the row of blocks is not shaded?

Write this fraction as a decimal.

Consider this diagram: a

What fraction of the grid is shaded?

Write this fraction as a decimal.

What fraction of the grid is not shaded?

Write this fraction as a decimal.

Write these fractions as decimals: a

Write these mixed numbers as decimals: a

Complete these conversion tables: a

Fraction Decimal

Decimal

0.1

Fraction

Decimal Ex 2

206

Fraction 0.5

Fraction Decimal

Write these fractions as decimals: a

Mathspace WA Curriculum – Year 7 mathspace.co

12 Ex 3

Ex 4

Ex 5

Write these fractions as decimals, rounding to the thousandths place, where needed: a

Write these mixed fractions as decimals, rounding to the thousandths place, where needed: a

Fill in the boxes with the missing numbers: a

Write these decimals as a simpliﬁed fraction: a

0.9

0.4

0.39

0.92

0.09

0.907

0.626

1.9

5.8

7.19

5.62

4.09

m 9.007

3.065

76.25

223.205

If 0.2 of a row of blocks is shaded: a

Find the decimal number that represents the unshaded part of the row.

Write the fraction that is unshaded in its simplest form.

If 0.72 of a grid is shaded: a

Find the decimal number that represents the unshaded part of the grid.

Write the fraction that is unshaded in its simplest form.

A recipe calls for make

times the recipe?

A container holds containers?

Reasoning 20

cups of ﬂour. How much ﬂour is needed in decimal form if you want to

of a litre of water. How much water is needed in decimal form to ﬁll 5

20, 21

21, 22

Identify which statements are true of the rectangle shown and explain why: a

More than half the bar is shaded

0.6 of the bar is unshaded

The bar is shaded between 0.6 and

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Can a fraction have a decimal equivalent that is greater than 1? Why or why not?

Answer these questions: a

Complete the table: Fraction Decimal

What pattern can you observe in the decimal values as the denominator gets larger? Explain.

Problem-solving 23a, 24a, 25 23

Place each set of numbers in descending order: a

23c, 24c, 26, 27

Place each set of numbers in ascending order: a

23b, 24b, 25, 26

A farmer has 3 plots of land on his farm. • On the first plot, which measures 0.6 hectares, he plants wheat on

of the area.

• On the second plot, which measures 0.9 hectares, he plants corn on

of the area.

• On the third plot, which measures 1.2 hectares, he plants soybeans on

of the area.

If he wants to plant sunflowers on the remaining land, how many hectares of land will be available for planting sunflowers? 26

A group of friends decided to split the cost of a meal. The first friend paid

of the total bill,

while the second friend paid 0.25 of the bill. The third friend paid the remaining amount. If the total bill was $120, how much did the third friend pay? 27

A recipe for a cake requires 1.5 cups of flour, cups of sugar and

cup of milk. A baker

decides to make a larger cake by doubling the quantity of flour and adding 0.75 cups of extra sugar. After these adjustments, how many cups of each ingredient does the baker need?

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5.04 Multiply decimals After this lesson, you will be able to… • use the vertical algorithm to multiply decimal numbers. • multiply with decimals using the area method.

Multiplication with decimals The vertical algorithm that we use to multiply whole numbers can also be used to multiply decimals. This method allows us to multiply in an organised way. Let’s look at the steps for multiplying decimals: 1. Align the decimal numbers vertically, with the larger number on top. 2. Ignore the decimal points and multiply the numbers as if they were whole numbers. 3. Count the total number of decimal places in both numbers being multiplied. 4. Place the decimal point in the product according to the total number of decimal places. Let’s say we want to multiply 4.83 by 5.7 using the vertical algorithm. We ﬁrst write the numbers one under the other, then we multiply the numbers as if they were whole numbers: 4 8 5 3 3 8 + 2 4 1 5 2 7 5 3 ×

3 7 1 0 1

Now we need to place the decimal point. To do so, we count the total number of decimal places in the original numbers. In this case, the original numbers are 4.83, which has two decimal places and 5.7, which has one decimal place. So their product will have 2 + 1 = 3 decimal places. Finally, we put the decimal point in the answer using the total number of decimal places, which is 3:

27.531 So we have found that 4.83 × 5.7 = 27.531.

Example 1 Evaluate 2.2 × 4

Create a strategy Multiply the numbers together as whole numbers and place the decimal point accordingly.

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209

Apply the idea Ignore the decimal point for the meantime and multiply the numbers together as whole numbers. ×

22 4 88

Add the total number of decimal places in the original numbers. Total number of decimal places = 1 + 0 =1

Add 1 for 2.2 and 0 for 4 Evaluate

We have a total of 1 decimal place. Take the product calculated earlier and insert the decimal point so that there is 1 decimal point. We now have: 2.2 × 4 = 8.8

Example 2 Evaluate 1.2 × 3.2, giving your answer as a decimal.

Create a strategy Multiply the numbers together as whole numbers and place the decimal point accordingly.

Apply the idea Ignore the decimal point for the meantime and multiply the numbers together as whole numbers. 32 12 64 + 320 384 ×

The result is 384. Add the total number of decimal places in the original numbers. Total number of decimal places = 1 + 1 =2

Add 1 for 1.2 and 1 for 3.2 Evaluate

We have a total of 2 decimal places. Take the product calculated earlier and insert the decimal point so that there are 2 decimal places. 1.2 × 3.2 = 3.84

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Idea summary To multiply two decimals together: • • •

Multiply the numbers together as whole numbers ﬁrst. Count the total number of decimal places in the original decimals. Place the decimal point in your answer so that it has the same number of decimal places.

The area method The area method is a visual way to multiply decimals that helps us understand what’s happening with the place value in our decimal numbers. It involves drawing a rectangle and dividing it into smaller parts to represent the decimals and whole numbers we’re multiplying. Let’s use the area method to multiply 3.4 × 34. 30

3 0.4

We multiply each pair of smaller rectangles and ﬁnd the area of each one. Then, we add up the areas of all the smaller rectangles to ﬁnd the total area, which is the product of the two numbers.

90 + 12 + 12 + 1.6 = 115.6

0.4

1.6

Find the total area

3.4 × 34 = 115.6

Example 3 In this question, we will use the area model to ﬁnd the product of a decimal and a whole number. a Find the area of each rectangle shown in the table as a decimal. ×

0.6

0.07

Create a strategy Multiply the sides of each rectangle to get the area.

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Apply the idea Area of 1st rectangle = 4 × 9 = 36 Area of 2nd rectangle = 6 × 9

Multiply the sides Evaluate Multiply the sides, ignoring the decimal point

= 54

Evaluate

= 5.4

Insert the decimal point

Area of 3rd rectangle = 7 × 9

Multiply the sides, ignoring the decimal point

= 63

Evaluate

= 0.63

Insert the decimal point

4 36

0.6 5.4

0.07 0.63

b Using the answer from part (a), ﬁnd 4.67 × 9 as a decimal.

Create a strategy Get the sum of the answers in part (a).

Apply the idea 4.67 × 9 = 36 + 5.4 + 0.63 We can work out this addition by computing vertically. If the total sum in any column is bigger than 9, carry the tens into the next column. 3 6 5 + 0 4 2

. 4 . 6 3 . 0 3

We now have: 4.67 × 9 = 42.03.

Idea summary We can multiply decimals using an area model. This allows us to break down the rectangle into sections to work out our problem in smaller steps. Then we can add our answers from each section to ﬁnd the total.

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5.04 Practice questions Understanding

Explorer 1, 2

Adventurer

Trailblazer

1–3

If you multiply two decimals that are less than 1, will the product be less than, equal to or greater than the decimals being multiplied? Why?

For these products, estimate if the answer will be: • Greater than both numbers. • Less than both numbers. • In between the two numbers.

3×5

3 × 0.5

0.3 × 0.5

7×4

0.7 × 4

0.7 × 0.4

15 × 8

15 × 0.8

1.5 × 0.8

What are the conditions such that the product of two decimal numbers is: a

In between the two numbers

Fluency 4

Ex 1, 2

Ex 3

Less than both of the two numbers

4e-l, 5g-l, 6, 7, 9, 11, 12, 14

4e-l, 5g-l, 6, 7, 9, 11–14

For these products: i

Identify the number of decimal places in the answer.

Calculate the answer.

0.3 × 0.5

0.2 × 0.54

0.1 × 0.2 × 0.3

0.8 × 0.05

0.03 × 0.05

0.007 × 0.006

0.9 × 0.5 × 0.02

0.015 × 0.2

4 × 0.009

64 × 0.001

18 × 0.07

38.5 × 0.01

4 × 50

4 × 0.5

0.4 × 0.5

Evaluate: a

4a-h, 5a-f, 6–10, 14

4×5

0.6 × 0.3

4.4 × 2

1.6 × 2.4

4.23 × 2

4.8 × 8

62 × 9.6

3.89 × 7

2.8 × 6.8

Determine whether the following statements are true or false: a

36 × 0.5 = 18 × 2

20 × 1.2 = 5 × 5

10 × 5 = 1.25 × 4 × 10

7.5 × 2 = 3 × 2 × 2.5

Consider a rectangle with side lengths of 2.34 and 5 units. a

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 2.34 × 5 as a decimal.

0.3

0.04

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Consider a rectangle with side lengths of 46.2 and 6 units. a

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 46.2 × 6 as a decimal.

Consider a rectangle with side lengths of 4.92 and 0.6 units. a

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 4.92 × 0.6 as a decimal.

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 2.3 × 3.1 as a decimal.

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 0.87 × 7.8 as a decimal.

0.2

0.9

0.02

0.3

0.8

0.07

3 0.1

Consider a rectangle with side lengths of 0.87 and 7.8 units. a

0.6

Consider a rectangle with side lengths of 2.3 and 3.1 units: a

40 6

7 0.8

Use the area method to ﬁnd 0.47 × 0.89 as a decimal.

Use the area method to ﬁnd 4.86 × 0.86 as a decimal.

Lisa purchased a car and is paying $148.40 each month to pay it off. If it takes her 60 months to pay it off, how much will she have spent on car repayments in total?

Reasoning

15, 16

16, 17

If 36 × 42 = 1512, what is the answer to 3.6 × 4.2? Why?

James used a calculator to evaluate 3.179 × 3.96 but forgot to enter the decimal points. The answer he got was 1 258 884. What should the answer have been? Why?

A bakery sells a loaf of bread for $3.50. Each loaf contains 20 slices.

214

If a customer buys 2 loaves of bread, how much do they have to pay in total?

Describe the steps to ﬁnd how much money the bakery would make if they sold 100 loaves.

If the price of each loaf of bread is increased by $0.25 and the bakery sells the same amount of loaves in a week, how much more money will they make?

Mathspace WA Curriculum – Year 7 mathspace.co

Problem-solving 18–21, 24 18

19–22, 24, 25

20–26

Find the area of a rectangle with these dimensions: a

Length of 0.5 cm, width of 0.8 cm

Length of 0.2 m, width of 0.07 m

A recipe calls for 0.75 cups of sugar and 0.5 cups of butter to make 12 cookies. If you want to make 36 cookies, how much sugar and butter will you need?

Neville trains six days a week. Each weekday, he runs 2.4 km and on Saturday, he runs 4.6 km. How far does he run in a whole week?

How much would you pay for eight t-shirts if they cost $46.88 each and you receive a discount of $9.47?

A store sells shirts for $23.99 each. If the customer buys a hat for $12.75 and also buys 4 shirts, what is the total cost of their purchase?

A company sells 2500 shares of stock for $19.35 per share. If the company then earns a further $18 765 on top of this, what is the new price per share, rounded to two decimal places, if the company distributes the profit evenly among all shares?

A local bakery sells two types of bread: whole wheat and sourdough. A loaf of whole wheat bread weighs 0.75 kg, while a loaf of sourdough bread weighs 1.25 kg. If the bakery sells 2.5 times more whole wheat bread than sourdough bread and has sold 80 loaves of whole wheat bread today, how many kilograms of bread have been sold in total?

Calculate the total weight of a shipment of boxes. There are both small and large boxes. Each box, regardless of size, weighs 1.25 kg and there are 3.5 times more small boxes than large boxes in the shipment. If there are 28 large boxes, how many small boxes are in the shipment and what is the total weight of all the boxes combined?

A factory produces rectangular metal sheets that are 0.5 metres wide and 1.75 metres long. Each sheet is then cut into smaller squares with side lengths of 0.25 metres. If the factory produces 3.5 times more sheets in a week than in a day and it produces 120 sheets in a day, how many small square pieces are produced in a week?

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5.05 Divide decimals After this lesson, you will be able to… • divide decimals by whole numbers. • divide decimals by decimals. • use short division to divide decimals by whole numbers and decimals.

Divide by a whole number Dividing with decimals uses a similar approach to dividing whole numbers. The only difference is that we work with the digits to the right of the decimal point. To evaluate division problems with decimals, we use short division. Sometimes we may need to rename some of our digits, just as we would when dividing whole numbers. The process of partitioning and renaming is what is used behind the scenes when we perform short division. Let’s take, for example, 92.4 ÷ 6. 6 goes into 9 once with 3 remaining, so we put a 1 in the tens column at the top and carry 3 to the ones column. 6 goes into 32 ﬁve times with 2 remaining, so we put a 5 in the ones column at the top and carry 2 to the tenths column. 6 goes into 24 four times with 0 remaining, so we put a decimal point and a 4 in the tenths column at the top. 92.4 ÷ 6 = 15.4

Example 1 Find the value of 8.904 ÷ 8.

Create a strategy Use short division.

Apply the idea Set up the short division. 8 goes into 8 once with 0 remaining, so we put a 1 in the ones column at the top.

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8 goes into 9 once with 1 remaining, so we copy the decimal point and put a 1 in the tenths column at the top and then carry 1 to the hundredths column. 8 goes into 10 once with 2 remaining, so we put a 1 in the hundredths column at the top and carry 2 into the thousandths column. 8 goes into 24 three times with 0 remaining, so we put a 3 in the thousandths column at the top. 8.904 ÷ 8 = 1.113

Idea summary We can use short division to divide a decimal by a whole number. It’s important to consider placement of the decimal point.

Divide by a decimal number We have already learned how to divide decimal numbers by whole numbers and it’s helpful to continue using this strategy when dividing numbers by decimals. For example, we know that 10 ÷ 5 = 2. But what is 100 ÷ 50? Well, that’s just also 2. Even though both numbers were ten times larger, we ended up with exactly the same answer. We can use the same strategy, but with numbers ten times smaller. This means 1 ÷ 0.5 = 2 as well. With this in mind, to divide any number by a decimal, we can ﬁrst multiply both numbers by a suitable power of 10 to make the divisor a whole number. This can make the division easier.

Example 2 Evaluate 1.2 ÷ 0.3

Create a strategy Multiply both numbers by a power of 10 to make the divisor a whole number.

Apply the idea 1.2 ÷ 0.3 = (1.2 × 10) ÷ (0.3 × 10)

Multiply both decimal numbers by 10

= 12 ÷ 3

Evaluate multiplication

Evaluate division

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Example 3 Evaluate 4.773 ÷ 0.03

Create a strategy Multiply by 100 to make the divisor a whole number. Proceed to short division.

Apply the idea Multiplying 0.03 and 4.773 by 100 will make the divisor 3 and the dividend 477.3. Using short division, 477.3 ÷ 3 = 159.1

Idea summary To divide a decimal by a decimal, we can ﬁrst multiply both numbers by a suitable power of 10 to make the divisor a whole number.

5.05 Practice questions Understanding 1

Give a whole number estimate for the answer to the division problem.

Choose the most reasonable estimate for 2.8 ÷ 7: A

Between 0 and 1

Between 1 and 7

Less than 0

Between 10 and 70

Choose the most reasonable estimate for 82.4 ÷ 4: 21

210

2.1

0.21

793.6 ÷ 8

56.231 ÷ 7

Nine ones

Nine tens

Estimate the value for these divisions: a

979.8 ÷ 2

Which statement is the same as 0.9? A

218

Trailblazer 1–3, 5

Write the problem shown in the short division in the form a ÷ b.

Adventurer 1–5

Explorer 1–5

Nine-tenths

7.776 ÷ 2

Nine

Find 0.9 ÷ 3, giving your answer as a decimal.

Mathspace WA Curriculum – Year 7 mathspace.co

Fill in the boxes to complete each equation: ⬚ ÷ 4.2 = 18.9

a c

⬚ = 18.9 ÷ 4.2

4.2 × 18.9 = ⬚ 4.2 × ⬚ = 18.9 ⬚ ÷ 5.6 = 25.2

⬚ = 25.2 ÷ 4.5

25.2 × 5.6 = ⬚ ⬚ × 4.5 = 25.2 6, 7aceg, 8, 9, 10a-ek, 11–13

Fluency 6

6, 7b-eg, 8, 9, 10a-ek, 11–13

6, 7e-h, 9, 10f-jl, 11–13

To ﬁnd 48.8 ÷ 4, we partition the number as follows: 48.8 = 40 + 8 + 0.8 a

Calculate: i

b Ex 1

Ex 2, 3

40 ÷ 4

8÷4

iii

0.8 ÷ 4

Add the answers to part (a) to ﬁnd 48.8 ÷ 4.

Evaluate: a

24.24 ÷ 3

11.54 ÷ 2

52.68 ÷ 6

245.37 ÷ 3

393.68 ÷ 7

588.6 ÷ 9

538.32 ÷ 6

1061.2 ÷ 8

Consider the quotient of 3.36 divided by 0.3 a

By what multiple of ten should both numbers be multiplied by, so that the divisor, 0.3, is a whole number?

Rewrite 3.36 ÷ 0.3 so that the divisor is a whole number.

Calculate 3.36 ÷ 0.3

Consider the quotient of 4.808 divided by 0.04 a

By what multiple of ten should both numbers be multiplied by, so that the divisor, 0.04, is a whole number?

Rewrite 4.808 ÷ 0.04 so that the divisor is a whole number.

Calculate 4.808 ÷ 0.04

Evaluate: a

56 ÷ 0.7

420 ÷ 0.7

1.8 ÷ 0.09

5.205 ÷ 0.05

12 ÷ 0.003

1.6 ÷ 0.4

8.1 ÷ 0.9

2.94 ÷ 0.07

84.06 ÷ 0.06

10.8 ÷ 0.012

33 ÷ 0.011

6.4 ÷ 0.008

On working days, Josie always buys a coffee on her way to work. Her coffee costs her $5.75 per day. One month, she budgets $120.25 for coffee. How many working days were there that month?

A bag of rice weighs 93.25 kg. A box contains 1585.25 kg of rice. How many bags are in the box?

Determine whether the following statements are true or false: a

9 ÷ 3 ÷ 3 < 9 ÷ 3 ÷ 2.7

12.6 ÷ 1.2 ÷ 5 < 12.6 ÷ 5 ÷ 1.2

15.2 ÷ 4 ÷ 0.8 < 16.2 ÷ 0.8 ÷ 4

9.2 ÷ 2.6 ÷ 1.3 < 6.5 ÷ 1.3 ÷ 2.6 5.05 Divide decimals mathspace.co

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Reasoning 14

14a-d, 15

14, 15

14, 16

Given that 235 × 42 = 9870, evaluate: a

9870 ÷ 42

9870 ÷ 235

98.70 ÷ 4.2

98 700 ÷ 0.42

0.987 ÷ 420

9.87 ÷ 0.235

Jo and Isla are asked to divide 1.6 ÷ 0.4. Jo multiplies both numbers by 10 before dividing. Isla multiplies both numbers by 1000. Will Jo and Isla get the same answer? Explain why or why not.

Casper is trying to determine the width of three rectangles. He has information about the area and the length of each rectangle, which are given to two decimal places. Without a calculator, he decides to round these measurements to the nearest whole number to estimate the width. a

The ﬁrst rectangle has an area of 19.73 square centimetres and a length of 4.23 cm.

Casper estimates the width using an area of 20 square centimetres and a length of 4 cm. He reasons that this will be an overestimate of the true width.

What answer did Casper get for the width?

Explain why his answer is an overestimate?

The second rectangle has an area of 30.22 square centimetres and a length of 5.83 cm. Casper estimates the width using an area of 30 square centimetres and a length of 6 cm. He reasons that this will be an underestimate of the true width. i

What answer did Casper get for the width?

Explain why his answer is an underestimate?

The third rectangle has an area of 39.65 square centimetres and a length of 7.91 cm.

Casper estimates the width using an area of 40 square centimetres and a length of 8 cm. He cannot decide if this will be an underestimate or an overestimate. i

What answer did Casper get for the width?

xplain why Casper has diﬃculty deciding whether this is an underestimate or an E overestimate?

Problem-solving 17–19

17–19

17, 19, 20

Juliet bought 2.2 kg of bananas for $8.60 and Luna bought 2.5 kg of bananas for $10.10. Who got the better value based on dollars per kilogram?

A water tank holds 20 L of water. Oliver is going to ﬁll the tank by ﬁlling a 600 mL bottle and emptying it into the tank. How many times will he need to do this?

A juice bar sells orange juice in 230 mL cups. One day they make 36.8 L of orange juice. How many cups could they sell?

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Lily makes metal window frames for a company that sells house and land packages. She makes two sizes of frames, one being 2.25 m wide by 1.5 m deep and the other being 2.25 m wide by 3.2 m deep. a

Calculate the perimeter of: i

the smaller frame.

the larger frame.

Lily currently has 150 m of metal for making frames. i

hat is the largest number of the smaller frames that she can make and how much W metal would she have left over?

hat is the largest number of the larger frames that she can make and how much W metal would she have left over?

iii

I f she wants to make the same number of each type of frame, what is the largest number of each type she could make and how much metal would be left?

5.06 Round decimals After this lesson, you will be able to… • round decimals to a place value. • round numbers to a specific number of places. • use estimation to approximate solutions and check the reasonableness of answers.

Round to a specific decimal place Rounding decimals is when we approximate a decimal number by shortening it to a specific number of decimal places. When indicating that a number is an approximation, we either use words such as, “rounded to a certain number of decimal places” or we use the approximation symbol, ≈. This can be particularly useful when dealing with money, measurements and calculations where exact values may not be practical. Let’s look at what it means to round a decimal to the nearest whole number.

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When we round a decimal number to the nearest whole number, we look at the digit to the right of the decimal point (tenths place). If this digit is 5, 6, 7, 8 or 9 we round up to the next whole number. If the digit is 0, 1, 2, 3 or 4, we round down to the current whole number. There are situations that require us to round numbers to a speciﬁc decimal place. For example, we round money to the nearest cent, length measurements to show accuracy to the nearest thousandths and our school marks to the nearest whole number or tenth. 5.06 Round decimals mathspace.co

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To round a decimal number to a speciﬁc decimal place, we need to follow these steps: 1. Identify the decimal place we want to round to. 2. Look at the digit to the right of the identiﬁed decimal place. 3. Apply the rounding rules.

Example 1 Round 2.19 to the nearest whole number.

Create a strategy Use a place value table to look at the digit to the right of the decimal point (tenths place). If this digit is 5 or greater, we round up to the next whole number. If the digit is less than 5, we round down to the current whole number.

Apply the idea Ones 2

. .

Tenths 1

Hundredths 9

The number in the tenths place is 1 which is less than 5, so we round down to the current whole number. The rounded answer is 2.

Reflect and check 2.19 is closer to 2 than 3. 2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

Example 2 Round 328.864 to the nearest hundredth.

Create a strategy Use a place value table to look at the digit to the right of the hundredths place. If this digit is 5 or greater, we round up to the next hundredth. If the digit is less than 5, we round down to the current hundredth.

Apply the idea Hundreds 3

Tens 2

Ones 8

. .

Tenths 8

Hundredths 6

Thousandths 4

The number in the thousandths place is 4, which is less than 5, so we round down and leave the digit in the hundredths place as 6. The rounded answer is 328.86.

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Idea summary To round a number to a speciﬁc decimal place: • Identify the decimal place we want to round to. • Look at the digit to the right of the desired decimal place. If this digit is less than 5, round down by dropping all digits to the right of the desired decimal place. If the digit is 5 or greater, round up by adding 1 to the desired decimal place and dropping all digits to the right.

Round to a number of decimal places Sometimes, instead of rounding to the nearest whole number, tenth, hundredth or thousandth, we might want to just specify how many digits we want to keep after the decimal point. When we want to round a decimal number to a speciﬁc number of decimal places, we follow these steps: 1. Identify the desired number of decimal places to keep. 2. Look at the digit to the right of the last desired decimal place. 3. Apply the rounding rules (rounding up if the digit is 5 or greater and rounding down if the digit is less than 5). To round 52.317 68 to three decimal places, Digit to the right

52.31768 ≈ 52.318 Digit being rounded

If we want to round a number to one decimal place, we are essentially rounding to the nearest tenth. This means we are looking at the ﬁrst digit after the decimal point. Similarly, if we want to round a number to two decimal places, we are rounding to the nearest hundredth, three decimal places is to the nearest thousandth, and so on.

Example 3 Round 79.38 to one decimal place.

Create a strategy Identify the ﬁrst number to the right of the decimal point and look at the next decimal place. Apply the rounding rules. Tens 7

Ones 9

. .

Tenths 3

Hundredths 8

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Apply the idea To round to one decimal place, we round the number at the tenths place, which is 3. We look at the number to the right (8) and apply the rounding rules. 8 is greater than 5, so we round up and add a 1 to the number in the ﬁrst decimal place: 3 + 1 = 4. We can round the value to 79.4.

Example 4 Round 1.599 64 to three decimal places.

Create a strategy

Apply the idea

We need to identify if 1.599 64 is closer to 1.599 or 1.600 as this is to 3 decimal places.

To determine if 1.599 64 is closer to 1.599 or 1.600, we look at the fourth decimal place. It is a 6. This is greater than 5, so we will round up. 1.599 64 will be 1.600 when rounded to 3 decimal places.

Idea summary Rounding to one decimal place is rounding to the nearest tenth, rounding to two decimal places is rounding to the nearest hundredth and rounding to three decimal places is rounding to the nearest thousandth and so on.

Estimation and reasonableness of solutions Estimation is a strategy that can be used to approximate a value or answer without solving the problem. By rounding numbers to the nearest whole number, we can make rough calculations and get an approximate answer that can help us check the reasonableness of our answer. For example, let’s say we want to estimate the product of 39.6 and 6.25. To estimate this, we can round 39.6 to the nearest whole number, which is 40 and round 6.25 to the nearest whole number, which is 6. Then, we can multiply these rounded numbers to get an estimate of the product: 40 × 6 = 240 This means that when we multiply 39.6 and 6.25, the answer should be near 240. If we get an answer that is signiﬁcantly different from 240, we know that we need to double-check our calculations.

Example 5 William earns $304.92 for 5 days of work. a What is the most reasonable estimate for how much William earns per day? A

224

$6.10

$610.00

Mathspace WA Curriculum – Year 7 mathspace.co

$61.00

$52.50

Create a strategy Round total earnings to the nearest dollar and divide this by the number of days.

Apply the idea Round $304.92 to $305. Divide this by 5, the number of days of work. 305 ÷ 5 = 61 The most reasonable estimate is Option C, $61.00.

b To the nearest cent, how much does William earn each day?

Create a strategy Use short division to ﬁnd the amount earned each day and round this up to the hundredths place.

Apply the idea Set up the short division of 304.92 by 5. Short division showing the quotient of 304.92 ÷ 5

5 goes into 30, so we put 6 in the tens column. 5 does not go into 4, so we put 0 in the ones place and carry 4 to the tenths place. Put the decimal point after the ones place. 5 goes into 49 nine times, so we put the 9 in the tenths place and carry the 4 to the 2 in the hundredths place. 5 goes into 42 eight times, so we put the 8 in the hundredths place and carry the 2 beside 0. 5 goes into 20 four times and so we put the 4 in the thousandths place with no remainder. To round 60.984 to the nearest cent is to round it to the hundredths place or to two decimal places. William earns $60.98 per day.

Reﬂect and check How does the actual answer compare to our estimate? Was our estimate within a reasonable range of the actual answer?

Idea summary Estimation is the process of ﬁnding an approximate value or answer to a problem without calculating the exact result. It helps us quickly check if our answer is reasonable.

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5.06 Practice questions Understanding

Is 11.34 closer to 11 or 12?

Is 38.892 closer to 38 or 39?

Which of these decimals are between 7 and 8?

8.201

4 and 5

Ex 3

226

7.39

5 and 6

6 and 7

7 and 8

Which of these pairs of consecutive whole numbers does 24.72 lie in between? 23 and 24

24 and 25

25 and 26

26 and 27

Based on your answer in part (a), what whole number is 24.72 closer to?

Fluency

8.85

Based on your answer in part (a), what whole number is 6.186 closer to?

Consider the number 24.72 a

7.99

Which of these pairs of consecutive whole numbers does 6.186 lie in between? A

Trailblazer 1–3, 5

Consider the number 6.186 a

Ex 2

1–4

Adventurer 1–3, 5

Ex 1

Explorer

6a-e, 7a-eg, 8a-e, 9a-f, 10a-eg, 11a-cf, 12ab, 13

6b-f, 7a-eg, 8a-e, 9a-f, 10ad-g, 11ad-f, 12a, 13, 14

6c-h, 7bd-h, 8e-h, 9c-egh, 10d-h, 11d-h, 12cd, 13, 14

Round these numbers to the nearest whole number: a

2.91

5.97

24.09

33.931

0.5

0.05

0.799

89.5

Round these numbers to the nearest tenth: a

0.14

7.35

3.56

8.99

27.88

289.427

0.7003

9.992

Round these numbers to the nearest hundredth: a

2.897

125.464

14.285

67.001

0.007

12.999

99.999

131.099

Round these numbers to one decimal place: a

0.41

8.35

19.38

0.3009

3.55

19.95

13.88

1.0099

Round these numbers to two decimal places: a

2.697

393.586

0.435

7.3582

79.997

2.0029

149.955

0.099

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 4

Round these numbers to three decimal places: a

11.8483

2.159 76

4.504 13

1.199 69

90.0335

0.5987

41.5005

9.9996

12.006 65

39.999 95

Round these numbers to four decimal places: a

Ex 5

5.659 64

What is the most reasonable estimate for how much Stephanie drives each day? A

10 km

50 km

100 km

Will this estimate be higher or lower than the actual average?

To the nearest km, what is the average kilometres travelled per day?

Trevor needs to read 564 pages of his book before it is due back at the library in 8 days. a

What is a good estimate for the number of pages he should read each day?

Will that estimate be higher or lower than the actual average? Explain how you know.

To the nearest whole page, what is the average number of pages Trevor should read each day to finish the book in time?

Reasoning 15

5.600 96

Stephanie drives a total of 756.25 kilometres over 7 days. a

15–17

Opie wants to write

15–17

16–18

rounded to three decimal places. Explain how many decimal places he

needs to calculate in order to do this correctly. 16

Josie wants to write 0.321 46 rounded to three decimal places. She says that the 5th decimal place, the 6, would cause the 4 in the 4th decimal place to become a 5 and then that would cause the 1 in the 3rd decimal place to become a 2. a

Explain what Josie has done wrong.

Round 0.321 46 to: i

Four decimal places.

Three decimal places.

Consider rounding 4.449. a

Round 4.449 to two decimal places.

If you were asked to round 4.449 to one decimal place, should you use the answer to part (a) or go back to the original number in the question? Explain.

5.43 is closer to 5.4 than it is to 5.5, so it would round to 5.4 and not 5.5 if we were rounding to one decimal place. 5.47 is closer to 5.5 than it is to 5.4, so it would round to 5.5 and not 5.4 if we were rounding to one decimal place. 5.45 is closer to exactly halfway between 5.4 and 5.5. What is the reason that we round it to 5.5?

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Problem-solving 19, 20 19

19, 20

19, 21

Jasper is calculating the perimeter of a rectangle. The rectangle is 8.79 cm long and 5.65 cm wide. a

Jasper does not have a calculator, so he decided to estimate the perimeter by rounding the length and width to the nearest whole number. What answer should he get for the perimeter?

Is his calculation an underestimate or an overestimate?

Calculate the perimeter of the rectangle to two decimal places.

What was the difference between Jasper’s estimate and the value calculated in part (c), rounded to one decimal place?

Tommy has $50 to buy a pair of trousers and a sweatshirt. The sweatshirt is advertised at $24.35 and the trousers at $26.25. To see if he has enough money, Tommy rounds both prices to the nearest dollar. a

What is Tommy’s estimate of the price of the trousers and the sweatshirt?

According to his estimate, does he have enough money?

Does he have enough money? Explain why or why not.

Is it appropriate to round down in this situation?

Juliet is trying to measure the weights of two bacterial cultures. The actual weights are 12.34 g and 15.78 g. However, the sensitive scale that she normally uses is broken and the one that she has only measures to the nearest gram. a

What are the weights measured by the scale that she can use?

What is the difference between the weights Juliet measured?

Is this an underestimate or an overestimate of the actual difference between the weights?

What is the actual difference between the weights, rounded to one decimal place?

Investigation: Account statements

Investigate online

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5 Chapter review 1

Which decimal is larger? 1.315 or 1.351

Write these list of numbers in descending order. 0.37, 0.073, 3.7, 0.0073, 0.000 73

Evaluate: a

0.5 + 0.2

0.45 + 0.29

3.314 + 1.686

120.317 + 879.683

0.12 + 0.92 + 0.67

25.4 + 42.06 + 12.005

Evaluate: a

7.8 − 6

6.2 − 3.1

12.5 − 5.8

0.65 − 0.42

9.63 − 4.3

8.47 − 1.6

0.672 − 0.245

95.36 − 42.15

Tyrone spends $4.50 at the grocery store. If he pays for it with a ten dollar note, how much change will he receive?

Write these fractions as decimals:

Write these decimals as a simplified fraction: a

0.001

0.37

0.12

0.625

4.6

3.68

8.012

59.75

On a scale drawing, a nail measures the drawing are

cm. In real life, we know that all the measurements on

times larger. What is the length of the nail in real life written as a decimal

to two decimal places? 9

Evaluate: a

0.4 × 0.7

0.02 × 0.06

0.3 × 0.6 × 0.03

5 × 0.008

1.8 × 2.2

4.35 × 3

75 × 8.3

3.21 × 6

Evaluate: a

36.36 ÷ 4

15.24 ÷ 3

68.34 ÷ 7

320.49 ÷ 4

481.44 ÷ 8

720.9 ÷ 9

675.84 ÷ 7

1360.8 ÷ 8

Evaluate: a

63 ÷ 0.9

2.4 ÷ 0.6

630 ÷ 1.05

6.15 ÷ 0.06

15 ÷ 0.003 75

12.96 ÷ 0.0144

36.3 ÷ 0.0121

7.2 ÷ 0.009

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On working days, Laura always buys a smoothie on her way to work. Her smoothie costs her $4.50 per day. One month she spent a total of $99.00 on smoothies. How many working days were there that month?

Round these numbers to four decimal places: a

3.245 73

7.831 28

10.056 94

22.999 52

Three packages with weights of 2.34 kg, 6.15 kg and 3.52 kg are to be shipped, but they exceed the shipping carrier’s 10.83 kg weight limit. By how much does the total weight exceed the limit?

Place each set of numbers in ascending order: a

A bookstore sells novels for $16.99 each. If the customer buys a bookmark for $3.50 and also buys 5 novels, what is the total cost of their purchase?

A recipe calls for 0.6 cups of cocoa powder and 0.4 cups of milk to make 10 servings of hot chocolate. If you want to make 30 servings of hot chocolate, how much cocoa powder and milk will you need?

Last year, a local bakery sold 1000 cupcakes at a price of $2.50 each, which only covered their costs, resulting in no profit. This year, the bakery aims to make a total profit of $4500 while maintaining the same production of 1000 cupcakes. What should be the new price per cupcake to achieve this goal?

A fish tank has a volume of 32 L. Sally is going to fill the tank by filling a 750 mL container and emptying it into the tank. How many times will she need to do this?

A candy store sells gummy bears in 120 g packets. One day they make 14.4 kg of gummy bears. How many packets could they sell?

Ally has $420 to buy a desk and a chair. The desk is advertised at $255.25 and the chair at $164.99. To see if she has enough money, Ally rounds both prices to the nearest dollar.

230

What is Ally’s estimate of the price of the desk and the chair?

According to her estimate, does she have enough money?

Does she actually have enough money?

Is it appropriate to round down in this situation?

Mathspace WA Curriculum – Year 7 mathspace.co

“The need for precision in measurement and calculation is the foundation of all scientific work.” Karl Pearson

Big ideas Percentages, like fractions and decimals, are a way of expressing a part of a whole or set. They provide a flexible system for dealing with real-world quantities, measurements and calculations.

Curriculum Outcomes • Determine percentages of quantities and express one quantity as a percentage of another using flexible and efficient strategies • Explore and explain relationships between fractions, decimals and percentages • Draw and label, or use a given number line, to locate, order and compare with equality and inequality symbols, fractions, terminating decimals, percentages and integers

6 Percentages Chapter outline 6.01 6.02 6.03 6.04 6.05

Fractions, decimals and percentages Compare percentages Express a quantity as a percentage Percentages of a quantity Percentages with technology Chapter 6 review

234 243 247 252 259 265

The word percentage comes from a Latin usage of per centum or per 100. Using the parts of some mathematical words can help you remember their definitions.

6.01 Fractions, decimals and percentages After this lesson, you will be able to… • define percent. • convert between fractions and percentages. • convert between decimals and percentages. • convert between fractions and decimals.

Meaning of percentage Percentage means ‘per hundred’. If we say 50% of something, it means Fractions

, which is the same as

Decimals

Percentages

1.0

100%

0.5

50%

or 0.5

These common fraction, decimal and percentage conversions are called benchmark values.

0.3 0.25

25%

0.2

20%

0.125 0.1

10%

0.05

0.04

Interactive exploration

Discover this concept in action online

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Idea summary Percentage means ‘per hundred’.

Convert percentages to decimals We can convert between decimals and percentages by taking advantage of the hundredths place value. 1% represents 1 hundredth or

or 1 ÷ 100, which we can write as 0.01 as a decimal.

We can convert any percentage to a decimal by dividing the percentage by 100, which is the same as decreasing the place value of each digit by two places and then removing the % symbol. For example, 83% represents 83 hundredths. This is 0.83 when written as a decimal. Similarly, 65.4% represents 65.4 hundredths. We can write this as the decimal 0.654.

Example 1 Write each percentage as a decimal: a 54%

Create a strategy To write a percentage as a decimal, divide by 100.

Apply the idea Divide by 100 and remove the % symbol

Write as a decimal

Reﬂect and check We wrote the percentage as a decimal representing the number of hundredths indicated by the percentage. Notice that converting the percentages to decimals had the same effect as decreasing the place value of the digits in the percentage by two places, then removing the % symbol. This is equivalent to dividing by 100 and removing the % symbol.

6.01 Fractions, decimals and percentages mathspace.co

235

b 125%

Create a strategy To convert into a decimal, divide by 100 and remove the % symbol.

Apply the idea Divide by 100 and remove the % symbol

Convert to a decimal

Idea summary We can convert any percentage to a decimal by dividing the percentage value by 100, which is the same as decreasing the place value of each digit by two places and removing the % symbol.

Convert decimals to percentages To convert from a decimal to a percentage, we can just reverse the steps from the previous section. We can convert any decimal to a percentage by multiplying the decimal by 100, which is the same as increasing the place value of each digit by two places and attaching a % symbol. Remember to attach the % symbol to the decimal at the same time as increasing the place values.

Example 2 Write each decimal as a percentage: a 0.09

Create a strategy To write a decimal as a percentage, multiply by 100%.

Apply the idea 0.09 = 0.09 × 100% = 9%

Multiply by 100% Evaluate

b 6.3

Create a strategy To write a decimal as a percentage, multiply by 100%.

Apply the idea 6.3 = 6.3 × 100% = 630%

236

Multiply by 100% Evaluate

Mathspace WA Curriculum – Year 7 mathspace.co

Idea summary We can convert any decimal into a percentage by multiplying the decimal by 100%, which is the same as increasing the place value of each digit by two places and attaching a % symbol.

Convert between percentages and fractions We can convert between fractions and percentages by remembering what percentage means. Percentage means ‘per hundred’ and any percentage can be written as a fraction with the denominator 100. To convert any fraction into a percentage, we can do the following: 1. Change the denominator of the fraction to 100 if possible and write as a percent. For example: 2. If the denominator isn’t easy to change to 100, such as , then convert the fraction to a decimal ﬁrst by dividing the numerator by the denominator. 3. Multiply the answer by 100 and then attach the % symbol. For example: 4. To check, use a calculator by pressing 3÷7, then ×100. After that, round the result to two decimal places and attach the % symbol. To convert any percentage into a fraction: 1. Write the percentage as a fraction with denominator of 100 and remove the % symbol. 2. If there is a decimal within the fraction, like in

, then multiply both the numerator and

denominator by a power of 10 so that the numerator becomes an integer, like

3. Simplify the fraction.

Example 3 Write each as a percentage: a

to two decimal places

Create a strategy To write as a percentage, convert to decimal first then multiply by 100%.

Apply the idea Convert to decimal

Multiply by 100%

Evaluate

6.01 Fractions, decimals and percentages mathspace.co

237

Reflect and check On a calculator, enter 4 ÷ 7 = to get 0.57142857142. Then multiply by 100 to get 57.1428571429. Then round it two decimal places to have 57.14%. b

Create a strategy To write a mixed number as a percentage, convert to an improper fraction first, then to decimal and lastly, multiply by 100%.

Apply the idea Convert to an improper fraction

Convert to decimal

Multiply by 100%

Evaluate

Example 4 Write each as a fraction in its simplest form. a 55%

Create a strategy To write a percentage as fraction, divide by 100.

Apply the idea Divide by 100 and remove the % symbol

Divide both the numerator and denominator by 5

Simplify the fraction

b 3.5%

Create a strategy To write a percentage as a fraction, divide by 100. There is a decimal within the percentage, so we need to multiply both the numerator and denominator by a number so that the numerator becomes an integer.

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Apply the idea Divide by 100 and remove the % symbol

Multiply both the numerator and denominator by 2

Simplify the fraction

Idea summary To convert any fraction into a percentage, we can do the following: 1. Change the denominator of the fraction to 100 if possible and write as a percent. For example: 2. If the denominator isn’t easy to change to 100, such as , then convert the fraction to decimal ﬁrst by dividing the numerator by the denominator. Then multiply by 100 attach the % symbol. For example: To convert any percentage into a fraction: 1. Write the percentage as a fraction with denominator of 100 and remove the % symbol. 2. If there is a decimal within the fraction like in

, then multiply both the

numerator and denominator by a power of 10 so that the numerator becomes an integer, like

3. Simplify the fraction.

6.01 Practice questions Understanding 1

Adventurer 1–5

Trailblazer 1–5

Complete these statements using the words ‘divide’ or ‘multiply’. a

Explorer 1–5

To convert a decimal into a percentage, you ⬚ by 100%.

To convert a percentage into a decimal, you ⬚ by 100%.

Complete this statement: The ﬁrst step in converting a percentage such as 45% into a fraction is to write it as a fraction with a denominator of ⬚. 15 out of the 30 class members were away ill. What percentage were away?

6.01 Fractions, decimals and percentages mathspace.co

239

Out of 100 students, 55 lived locally. What percentage did not live locally?

Complete each equation in their simplest forms:

c 6

What does it mean when a percentage is more than 100%?

Fluency

7ace, 8a-d, 9a-d, 10a-d, 11ab, 12a-d, 13ac-e, 16, 17

7a-c, 8ce-g, 9acef, 10e-h, 11a-d, 12b-df, 13bf-h, 14a-c, 16, 17

7b-d, 8cdgh, 9cdgh, 10i-l, 11cd, 12d-h, 14-17

For each of these number patterns: i

Rewrite the pattern as percentages.

Rewrite the pattern as decimals.

e Ex 1

Ex 2

10 Ex 3

Write each percentage as a decimal: a

18%

57%

23.4%

5.5%

120%

303%

500%

6897%

Write each decimal as a percentage: a

0.07

0.857

5.3

13.04

0.041

0.0035

1.375

Express each fraction as a percentage, rounding your answer to two decimal places where necessary: a

Express each fraction as a percentage: a

240

Express each percentage as a decimal, rounding your answer to three decimal places where necessary: a

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 4a

Ex 4b

Write each percentage as a fraction in its simplest form: a

75%

40%

10%

13%

68%

36%

Write each percentage as a fraction or a mixed number in its simplest form: a

7.5%

2.5%

0.2%

68.5%

109%

450%

360%

285%

Convert between decimals, percentages and fractions to complete the table, expressing the fractions in simplest form:

Decimal

Percentage

Fraction

0.53

0.6%

430% 2.8 11.25 16

Convert each as a percentage: a

4.5

e 17

9.04

13.82

Convert each fraction to a percentage rounded to four decimal places: b

Reasoning

3.67

18, 22, 23

18, 19, 22, 23

19–23

Josie heard an athlete say that he was going to give 110% in the ﬁnal. She says this is impossible because you cannot have more than 100%. a

Is Josie correct in the context of her statement?

Is Josie’s statement correct in the context of a test result?

Can you think of a context in which what Josie said was wrong and where you could get more than 100%?

Arrange the following numbers in ascending order: 0.24,

6.01 Fractions, decimals and percentages mathspace.co

241

In the decimals shown in the table, x, y and z represent different digits. A x. yz

B 0.0zx

C 0. xyz

D 0. zyx

Match each percentage below to a decimal equivalent in the table: a 21

xy. z%

xyz%

Suppose you were converting

zy. x%

z. x%

to a percentage. Describe a strategy for doing this that

involves ﬁnding equivalent fractions with denominators that are powers of 10. 22

Jasper won a match in a table tennis competition. The winner of a match was the ﬁrst player to win 4 games and the winner of each game was the ﬁrst player to win 11 points. The scores were 11 − 7, 8 − 11, 11 − 5, 11 − 4, 2 − 11, 11 − 8. a

What fraction of the games did Jasper win?

How many points were played in total?

What fraction of the points did Jasper win?

What percentage of the points did Jasper lose?

A shop advertises a “180% increase” in customers. Is this the same as saying the number of customers tripled? Justify your answer.

Problem-solving 24–26, 30

24–26, 28, 30

24–30

In a survey, 75% of the respondents said they prefer coffee over tea. 0.1 of the respondents expressed no preference. What percentage prefer tea over coffee?

A bag of toffee has 50 pieces in it. If 15 pieces have been eaten, what percentage of the original pieces of toffee remain?

In a game, a player scored 24 out of the 30 points scored by her team. Express the points scored by the rest of her team as: a

A simplified fraction

A percentage

A decimal

A bag has a variety of coloured marbles. It is known that 30% of the marbles are red and that there are 87 red marbles. How many marbles are there in the bag that are not red?

On her ﬁrst Mathematics test, Juliet scored 16 out of 25. On the second Mathematics test, she scored 27 out of 40. On which of the tests did she get the higher percentage?

A 5-a-side soccer team has a 20 match season. After 16 games, they had won 12 matches. a

What is the largest possible win percentage they could achieve for the season?

What is the smallest possible win percentage they could achieve for the season?

A recipe needs 0.9 of a litre of milk. If you triple the recipe, what percentage of a litre of milk do you need?

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A toy costs $20. After a sale increases its price by 225%, what is the new price?

Jennie’s entire villa is composed of her house which is the biggest, a recreational area and a greenhouse which is the smallest. The house is of the recreational area and the greenhouse.

of the entire villa. Find the percentages

6.02 Compare percentages After this lesson, you will be able to… • convert between fractions, decimals and percentages. • define and use benchmark conversions for fractions, decimals and percentages. • compare and order numbers presented in different formats.

Compare fractions, decimals, percentage When comparing values that have different representations, like fractions, decimals or percentages, one common strategy is to convert them all to the same representation. Arranging numbers in ascending order means smallest to largest; descending order means largest to smallest.

Example 1 In this question, we will be working with the numbers , 60% and 0.3. a Convert

to a percentage.

Create a strategy Write as a fraction with a denominator of 100 first, then as a percent.

Apply the idea

Write as an equivalent fraction with a denominator of 100 Write as a percentage

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b Convert 0.3 into a percentage.

Create a strategy To convert a decimal to a percentage, we need to write the decimal as hundredths.

Apply the idea Write as hundredths

Write as fraction

Write as percentage

c Write , 60% and 0.3 from largest to smallest.

Create a strategy Compare the percentages.

Apply the idea From part (a), From part (b), 0.3 = 30% This means that from largest to smallest, the percentages are: 60%, 30%, 25% Writing the list using the original numbers is: 60%, 0.3,

Idea summary Ascending order means smallest to largest; descending order means largest to smallest. To compare decimals, percentages and fractions, convert them to the same form, then compare.

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6.02 Practice questions Understanding 1

Explorer

Adventurer

1–3

2, 3

Trailblazer 2, 3

What fraction of squares are shaded in this grid?

What percentage of squares are shaded?

What fraction of triangles are shaded in this pentagon?

What percentage of triangles are shaded?

James is offered three deals after an Easter egg hunt: • Deal 1: He keeps 12% of the total eggs found. • Deal 2: He keeps 0.65 of the total eggs found. • Deal 3: He keeps

of the total eggs found.

How can James work out which deal to take? 4ab, 5bd, 6a-d, 7, 8a, 9ac

Fluency 4 Ex 1

Rewrite the group as percentages.

Order the original group from smallest to largest. b

Identify whether these values are the same or determine which number is larger: a

4cd, 5ac, 6e-h, 8b-d, 9d

For these groups of numbers:

a 5

4bc, 5bd, 6e-h, 7, 8b, 9a-c

Determine which inequality sign, < or >, would make the statement true: a e

69% ⬚ 0.65 ⬚ 131%

b f

71% ⬚ 0.59 ⬚ 62%

c g

51% ⬚ 0.87 ⬚ 0.88

d h

0.052 ⬚ 4.9% 0.095 ⬚

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Are these statements true or false? a

91% > 0.91

63.2% < 0.72

Arrange each of these sets of numbers in descending order:

a c

92%, 0.1, 0.36, 60.1%

Reasoning

11–13

10–12

10, 12, 13

Write 3 different percentages that lie between

Write 3 different decimals that lie between 69.2% and 69.4%.

Write 3 different fractions that lie between 75% and 80%.

Write 3 different integer values for a and b so that

Problem-solving 14–16 14

and

is equal to 68%.

14–16

14, 16, 17

Consider the number line with five points labelled a, b, c, d and e: de

a 1

Match each letter to the correct value:

Determine which inequality sign, < or >, would make the statement true: i

c 15

iii

Arrange the numbers in descending order.

Josie, Juliet and Jasper had a box of chocolates to share. Josie ate

of the contents of the

box, Juliet ate 30% of the contents of the box and Jasper ate 0.15 of the contents. There were 12 chocolates left in the box. How many chocolates were in the box originally?

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Bernie wants to buy a laptop. It is priced at $1500 and is being offered on sale at 3 different shops. Shop A is offering 25% off, Shop B is offering

off and Shop C says that the purchase

price is 0.7 of the normal price. Which shop is offering the best discount? 17

Romeo, Ollie and Stella won some money in the lottery and decided to share it in proportion to the amounts that they contributed towards the purchase of the tickets. Romeo had paid of the purchase price and Ollie had paid 0.28 of the purchase price. Who should get the largest share of the winnings?

6.03 Express a quantity as a percentage After this lesson, you will be able to… • find parts of a whole. • express a quantity as a percentage.

Quantities as percentages When finding quantities as percentages, we always want to compare our quantity to the whole that it is a part of. When writing this as a fraction, we always put the quantity as the numerator and the whole as the denominator. Once we have our quantity written as a fraction of the whole, all we need to do is convert it into a percentage, by multiplying by 100 and adding the percentage sign.

Example 1 What is 27 as a percentage of 150?

Create a strategy Write the quantity as a fraction of the whole, then multiply by 100%.

Apply the idea 27 as a percentage of 150 is equal to

. Multiply by 100%

Evaluate

Simplify

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Example 2 There are 2 boys and 7 girls in a class. a Find the total number of students in the class.

Create a strategy

Apply the idea Total = 2 + 7

Add the number of boys and girls.

Substitute values Evaluate

b What percentage of the class are boys? Round your answer to two decimal places.

Create a strategy

Apply the idea

Write down the number of boys as a fraction of the total number of students in the class then multiply by 100% to convert it to a percentage.

Multiply by 100%

Evaluate

c What percentage of the class are girls? Round your answer to two decimal places.

Create a strategy Subtract the percentage of boys from 100%.

Apply the idea Percentage = 100% − 22.22% = 77.78%

Subtract the percentages Evaluate

Idea summary To find a quantity as a percentage, first write it as a fraction with the quantity as the numerator and the whole or the total as the denominator, then multiply the fraction by 100%.

Quantities with different units Sometimes when writing a quantity as a percentage, the quantity may not be in the same units as the whole we are comparing it to. In order to solve these problems, we ﬁrst need to convert to the same units so we can set up the fraction correctly.

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Example 3 What percentage of 4 hours is 36 minutes?

Create a strategy We must ﬁrst convert the times to the same unit.

Apply the idea Remember that 1 hour = 60 minutes. Multiply by 60

Evaluate

Multiply by 100%

Evaluate

Simplify

Idea summary To compare quantities with different units, we need to convert one of the quantities so it is in the same units as the other.

6.03 Practice questions Understanding 1

Adventurer 1bdf, 2

Trailblazer 1f, 2

Complete these statements: a b c d e f

Explorer 1ace, 2

30% is the same as ⬚ out of 100 and ⬚ out of 10.

15% is the same as ⬚ out of 100 and ⬚ out of 20. 75% is the same as ⬚ out of 100 and ⬚ out of 4.

80% is the same as ⬚ out of 100 and ⬚ out of 5.

50% is the same as ⬚ out of 100 and ⬚ out of 2.

62% is the same as ⬚ out of 100 and ⬚ out of 50.

Which of these shows how to calculate 30 as a percentage of 65? A

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3aceg, 4ae, 6a, 7ace, 8-10, 11aceg

Fluency 3

With a denominator of 100.

In simplest form.

40%

65%

30%

16%

25%

62.5%

75%

35%

17 out of 6

Express these quantities as a: i

Fraction

Percentage

21 out of 50

36 out of 45

4 out of 80

28 out of 56

42 out of 24

44 out of 110

64 out of 40

21 out of 175

What percentage of the total triangles are shaded? i

iii

What percentage of the total squares are shaded? i

What percentage of one whole circle is shaded in each set? i

Write these quantities as percentages: a

250

3dfh, 4bfgh, 5, 6cd, 7bdf, 8, 10, 11bdfh

Express these percentages as a fraction: i

3a-d, 4ac-e, 5, 6ab, 7ace, 8, 10, 11bdfh

36 out of 200

10 out of 16

Mathspace WA Curriculum – Year 7 mathspace.co

9 out of 27

Ex 1

Find: a

30 as a percentage of 150

81 as a percentage of 150

34 as a percentage of 56

12 as a percentage of 32

6 as a percentage of 60

8 as a percentage of 12

Calculate 28 cm as a percentage of 3 m.

A class has 32 students and 12 of them are left-handed.

Ex 2

How many of the students in the class were not left-handed?

What percentage of the class is left-handed?

What percentage of the class is not left-handed?

What percentage of 2 hours is 48 minutes?

Ex 3

Find: a

443.22 km as a percentage of 890 km

164 cm as a percentage of 8.2 m

$6 as a percentage of $375

$966 as a percentage of $27 600

45.5 kg as a percentage of 65 kg

8550 mL as a percentage of 28.5 L

54 min as a percentage of 6 hrs

108.9 mL as a percentage of 330 mL

Reasoning

12, 14

13, 14

13, 15

In a pond, there are 35 ducks, 12 geese and 10 swans. What percentage of the birds in the lake are ducks? Give your answer to two decimal places.

A class has g girls and b boys.

Write an expression for the percentage of girls in the room.

Write an expression for the percentage of boys in the room.

Consider these sales: • Sale 1: A jacket normally priced at $120 was on sale for $102 • Sale 2: A pair of trousers normally priced at $78 was on sale for $65 Which of these represents the largest percentage reduction?

A group of p people attended a conference; q of them were from the United States. a

Write an expression for the percentage of attendees who were from the United States.

Write an expression for the percentage of attendees who were not from the United States.

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Problem-solving 17

16, 18, 19

16, 18, 20

A car weighs 1650 kg and a truck weighs 3 tonnes. Express the mass of the car as a percentage of the mass of the truck.

A box of fruit contains 8 apples, 8 mandarins and 4 pineapples. a

Find the total number of pieces of fruit inside the box.

What percentage of the box are apples?

What percentage of the box are mandarins?

What percentage of the box are pineapples?

In a factory producing light bulbs, it is found that 27 out of a batch of 300 light bulbs are faulty. What percentage of the light bulbs are faulty?

Deborah scored 40 out of 50 in her history test and 57 out of 75 in her mathematics test.

What percentage score did Deborah get for her history test?

What percentage score did Deborah get for her mathematics test?

On which test did Deborah perform better?

Justin and Georgia handed out flyers to advertise for their new concert. Justin handed out 120 of his 400 flyers while Georgia handed out 360 of her 600 flyers. a

What percentage of his flyers did Justin hand out?

What percentage of her flyers did Georgia hand out?

Who handed out a higher percentage of their flyers?

6.04 Percentages of a quantity After this lesson, you will be able to… • find percentages of quantities using different strategies.

Find percentages by using smaller percentages We know that we can use fractions to express parts of a whole. We can use this to help us ﬁnd percentages of quantities.

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The benchmark values table gives us fractions that we can use to ﬁnd percentages: Fractions

Decimals

Percentages

1.0

100%

0.5

50%

0.25

25%

0.2

20%

0.125

12.5%

0.1

10%

0.05

0.04

We can see from the table that 100% is a whole, 50% is a half and 25% is a quarter. If we want to ﬁnd 50% of 30, we can ﬁnd half of 30, which is 15. It’s useful to know that 10% means

or ‘divide by ten’

For example, if we want to ﬁnd 15% of 120: 10% of 120 is 12. To get 5% of 120, find half of 12 which is 6. Then add to get 15%. So 15% of 120 is 12 + 6 which is 18.

Example 1 Suppose we want to ﬁnd 28% of a quantity. a Which of these statements is the same as 28%? A

Seven groups of 100%

Twenty eight groups of 100%

Seven groups of 4%

Eight groups of 4%

Create a strategy

Apply the idea

Work out the total percentage for each option.

Seven groups of 100% equals 700%. Twenty eight groups of 100% equals 2800%. Seven groups of 4% equals 28%. Eight groups of 4% equals 32%. So the correct answer is option C

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b What is 4% of 175?

Create a strategy

Apply the idea

To ﬁnd 4% of 175, divide it by 25.

Divide by 25

Evaluate

c Find 28% of 175.

Create a strategy Combine the information from parts (a) and (b) to solve.

Apply the idea In part (a), we found that 28% is equal to 7 groups of 4%. In part (b), we found that 4% of 175 is equal to 7. Combining our information from parts (a) and (b), we ﬁnd that 28% is equal to 7 groups of 7. 28% of 175 = 7 × 7 = 49

Multiply by the amount of groups Evaluate

Idea summary If we can write the percentage we want to find as some multiple of a smaller percentage, we first find that smaller percentage. Then, we multiply the result by the number of smaller quantities that fit into the original percentage. It’s useful to know that 10% means ‘divide by 10’.

Find percentages directly In the case where we can’t break up a percentage into smaller, easier to find pieces, we can calculate the percentage directly. We can do this by converting our percentage into a fraction or decimal and applying that directly to the quantity. For example, we can write 48% of 60 as

Example 2 What is 23% of 16? Write your answer as a decimal.

Create a strategy Convert the percentage into a decimal then multiply it by the given quantity.

Apply the idea 23% of 16 = 0.23 × 16 = 3.68

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Convert to a decimal and multiply Evaluate

Mathspace WA Curriculum – Year 7 mathspace.co

Example 3 We can find 15% of a quantity by first converting the percentage to a fraction. a Write 15% as a fraction in its simplest form.

Create a strategy To write a percentage as a fraction, divide by 100.

Apply the idea Write as a fraction with denominator 100

Divide both the numerator and denominator by 5

Simplify the fraction

b Find 15% of 160.

Create a strategy Multiply the fraction found from part (a) by the given quantity.

Apply the idea Multiply the fraction by the given quantity

Evaluate

Simplify the fraction

Idea summary We can ﬁnd percentages of quantities directly by multiplying the quantity by either the fraction or decimal equivalent to the percentage.

Percentages greater than 100% So far, the percentages that we have found have always been below 100%. As a result, the solution has always been smaller than the original quantity. This is because we have been taking a fraction of the whole. However, if the percentage we are looking for is greater than 100%, then we are taking more than one whole and we should end up with more than our original quantity.

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For example: 200% of $45 means two lots of 100% which is 2 × $45 = $90. Notice that $90 is a larger number than $45. When ﬁnding a percentage of a quantity, swapping the values of the percentage and the quantity does not affect the ﬁnal solution.

Example 4 What is 350% of 16?

Create a strategy Break down the percentage into smaller multiples then multiply by the given quantity.

Apply the idea Notice that 350% is equal to 3 lots of 100% and 1 lot of 50%. 350% of 16 = 3 × 100% + 1 × 50% = 3 × 16 + 1 × 8 Replace the 100% with our amount of 16 and 50% with half of 16 which is 8. = 48 + 8

Evaluate the multiplication

= 56

Evaluate

Idea summary If the percentage we are looking for is greater than 100% then we are taking more than one whole. Remember that we should end up with more than our original quantity.

6.04 Practice questions Understanding 1

Explorer 1–4

Adventurer 1–4

Trailblazer 1–4

Complete the table of common percentages and their equivalent fractions. Fraction Percentage

10%

Write the fraction or number that completes these statements: a b

256

50%

25% of a quantity is the same as ⬚ of that quantity.

200% of a quantity is the same as ⬚ of that quantity.

Mathspace WA Curriculum – Year 7 mathspace.co

What number do you divide by to ﬁnd 10% of a quantity?

Explain what it means if a percentage of a quantity is more than 100%. 5a-c, 6, 7, 8a-cl, 9a-ce-g, 10ace

Fluency Ex 1

5–7, 8degk, 9ace-h, 10ace

5–7, 8fh-j, 9cd, 10bdf

Follow the steps in each case to determine the given quantity: a

75% of 28 i

How many groups of 25% go into 75%?

What is 25% of 28?

What is 10% of 35?

What is 4% of 225?

What is 1% of 300?

iii Find 75% of 28. b

140% of 35 i

How many groups of 10% go into 140%?

iii Find 140% of 35. c

28% of 225 i

How many groups of 4% go into 28%?

iii Find 28% of 225. d

7% of 300 i

How many groups of 1% go into 7%?

iii Find 7% of 300. 6 Ex 2

7 Ex 3

Ex 4

Determine the value of these quantities by first converting the percentage to a decimal: a

90% of 20

220% of 10

60% of 60

21% of 40

Determine the value of these quantities by first converting the percentage to a fraction: a

15% of 200

250% of 12

16% of 275

175% of 20

Evaluate the percentages without using a calculator: a

10% of 90

25% of 16

1% of 900

2% of 400

500% of 11

350% of 28

60% of 30

45% of 160

200% of 13

150% of 18

80% of 40

50% of 320

Find: a

10% of 80 g

20% of $60

30% of $200

2% of 300 mL

90% of 80 kg

70% of 20 km

25% of 120 L

75% of $24

Which is greater? a

62% of 38

62% of 46

27% of 230 or 13% of 230

70% of 30

250% of 18

45% of 180 or

13% of 17

29% of 13

36% of 48 or 24% of 63

7% of 700

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Reasoning

11, 14

11, 12, 14

13, 14

Show two different ways to correctly calculate 25% of 140.

a b c

hen finding a percentage of 82 that is above 100%, explain whether the result will be W smaller or larger than 82. When finding a percentage of 82 that is below 100%, explain whether the result will be smaller or larger than 82. Show two different ways to find: i

60% of 82

120% of 82

Opie mowed 40% of his lawn in 36 minutes. How long would it take him to mow the whole lawn if he continues to mow at the same constant rate?

a b c d

Calculate 40% of 70. Calculate 70% of 40. What do you notice about 40% of 70 and 70% of 40? Does this result apply in general? Investigate by writing expressions for calculating p% of q and q% of p.

Problem-solving 17–20

15, 17, 18, 20

15–17, 19, 21

Josie calculated 180 ÷ 9 ÷ 4 × 6. What percentage of 180 did she produce? Give your answer to two decimal places.

A water tank is filled to 15% of its capacity in 10 minutes. How long will it take to fill the tank if, after the first 10 minutes, the rate of flow is double than it was in the first 10 minutes? Give your answer in minutes and seconds.

Lisa scored 95% on her geography exam, which was marked out of 160. What was Lisa’s actual mark out of 160?

A theatre has a capacity of 3000 people. On the opening night, the theatre was 97% full. On the second night, it was 78% full.

258

How many people attended the theatre on the ﬁrst night?

How many people attended the theatre on the second night?

How many people attended the theatre in total during the ﬁrst 2 nights?

A stadium can hold 5000 people. The stadium is 70% full and 11% of the audience are children. a

How many people are in the stadium?

How many children are in the stadium?

Mathspace WA Curriculum – Year 7 mathspace.co

Yuri is eating a box of cookies. The nutritional information states that each cookie contains 7% of his recommended daily intake of sugar. If his recommended daily intake of sugar is 37 500 mg, how many milligrams of sugar does each cookie contain?

Kiki’s salary is $96 000 per year. She is given a pay rise of 4%. a What is the amount of the pay rise? b What would her new annual salary be? c How much would she be paid per week at the new salary level? d What percentage rise would she have needed to give her a salary of $100 000? Give your answer to two decimal places.

6.05 Percentages with technology After this lesson, you will be able to… • use technology to perform mathematical operations and solve percentage problems.

Percentages and decimals using technology We can use a calculator to evaluate expressions with fractions, decimals or percentages.

S D

On most handheld calculators, we can use this button to convert between a fraction and a decimal.

When converting percentage values into decimals on the calculator, divide by 100. E.g. 83% as a decimal is 83 ÷ 100 = 0.83. Some calculators have a percentage function which has the same effect as dividing by 100. It should be noted that the calculator cannot give percentage values as an output; it will usually show the answer as a decimal. If your final answer needs to be a percentage, then multiply your decimal value by 100 and add the % symbol manually to your written answer.

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Example 1 Use a calculator to convert

to a decimal.

Create a strategy Use a calculator to convert the fraction to decimal. 1. Use the

button to enter a fraction.

2. Enter the numerator and denominator.

Apply the idea Convert to a decimal

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Example 2 Use a calculator to evaluate 51% × 83%. Write the answer as a decimal rounded to three decimal places.

Create a strategy Using a calculator, enter the expression in the input bar to evaluate the expression.

Apply the idea 51% × 83% = 0.423

Evaluate

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Example 3 Lisa scored 70% on her Maths exams which has a total of 140 items. What was Lisa’s actual mark out of 140?

Create a strategy Multiply the total marks by the percentage. Use a calculator to ﬁnd the value of the actual mark.

Apply the idea Mark = 70% × 140 = 98

Multiply the percentage by the total marks Evaluate

Idea summary Calculators can be used to quickly and accurately perform calculations involving percentages and decimals.

6.05 Practice questions Understanding 1

Adventurer 1, 2ac

Trailblazer 1, 2cd

Complete these steps to express 45% as a fraction or decimal using a calculator. a b c

262

Explorer 1, 2ab

Press ⬚ into the calculator.

Then, press ⬚ to show the equivalent fraction or decimal.

Lastly, press ⬚ to convert the value from fraction to decimal or decimal to fraction.

Mathspace WA Curriculum – Year 7 mathspace.co

Describe the process on how to find the values of each statement. You do not need to find the actual answer. a

29 out of 60

40% of 12

130.8% of 9

The number where 17 is its 80%

Fluency 3

3bceh, 4, 5bcfh, 6a-c, 8–10

Ex 1

5 Ex 2

8 Ex 3

44 out of 64

c 899 out of 725

d 5.9 out of 0.32

76 out of 91.2

Use a calculator to convert these to decimals: b

Use a calculator to convert these to decimals: a e

3adeg, 4, 5adeg, 6bdef, 7

Use a calculator to express these as percentages: a

3bcef, 4, 5bdeg, 6a-c, 9, 10

b 135% × 24.6

9.25%

53% × 85%

70% of 617

of 65.86

72% ÷ 3

126% of 87.2

Use a calculator to find: a

23% of

91.6% of 97.3

209% of

47.3% of 1.764

317% of

Use a calculator to determine the value of x: a

70% of x is equal to 42.7

60% of x is equal to 39.54

60.1% of x is equal to 24.04

46% of x is equal to 35.88

Joanne scored 80% on her Maths exam, which was marked out of 160. What was Joanne’s actual mark out of 160?

A city has a yearly water supply of 27 000 000 megalitres. It gets 14% of this water supply from melted snow. Find the amount of water that comes from melted snow each year.

Ryan scored 60% on his Maths exam, which was marked out of 110. What was Ryan’s actual mark out of 110?

Reasoning 11

11, 12

12, 13

Give some examples of when it might be quicker not to use a calculator in percentage calculations.

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Bernie works at a clothing store. He buys a jacket that is 25% off in the sale. He also receives a staff discount of 12%. a

To calculate the price he needs to pay, he multiplies the original cost of the jacket by 0.75 and then by 0.88. Explain his calculation.

Betty tells Bernie that 0.75 × 0.88 = 0.66; so he could have just multiplied the price of the jacket by 0.66. Verify that Betty is correct.

Would Bernie get the same result if he applied the two discounts in the reverse order, i.e. a discount of 12% followed by a discount of 25%?

Casper got a promotion at work. His new post meant that his salary increased by 8%. He then received his annual cost of living rise of 4%. a

Casper reasons that since 8% + 4% = 12%, he should multiply his original salary by 1.12. Will this give him the correct value for his new salary?

Casper’s manager tells him that he is doing the calculation incorrectly. She suggests that he should multiply the original salary by 1.1232. Explain how she arrived at this ﬁgure.

Who has the correct calculation, Casper or his manager?

Would Casper be on the same new salary if the cost of living rise of 4% was applied before the promotional rise of 8%?

Assuming that his manager’s calculation is correct, what is Casper’s salary after both increases if his original salary was $72 000? Give your answer to the nearest dollar.

Assuming that his manager’s calculation is correct, what was Casper’s original salary if, after both increases, his salary was $89 856? Give your answer to the nearest dollar.

Problem-solving 15 14

264

14, 15

14–17

A bag of potatoes weighs 50 kg. When more potatoes are added, the weight of the bag increases by 35%. a

Find the added weight in the bag of potatoes.

Find the new weight of the bag of potatoes.

Duncan is going to buy a hat that is marked as 25% off. The original price was $30. a

Find the discount amount.

Find the discounted price that Duncan will pay for the hat.

The 64 girls at Victoria’s party represent 80% of all the guests. a

How many guests represent 10% of all the guests?

Find the total number (100%) of guests at the party.

Jane paid $750 to get a dress made. $450 of this was for the materials and the rest was for the labour required. a

Find the percentage of the cost that paid for the materials.

Find the percentage of the cost that paid for the labour.

Mathspace WA Curriculum – Year 7 mathspace.co

6 Chapter review 1

Write each percentages as decimals: a

10%

93%

23.4%

4.5%

160%

409%

700%

7894%

Write each decimals as percentages: a

0.03

0.021

0.457

6.2

15.04

0.0035

3.325

Express each fraction as percentages: b

a 4

Convert between decimals, percentages and fractions to complete the table, expressing the fractions in simplest form:

d Fraction

Decimal

Percentage

1.51

Express each percentages as both a fraction and a decimal: a

20%

Rewrite the group as percentages.

Order the original group from smallest to largest. b

For each of these expressions, determine which inequality sign, < or >, would make the statement true: a

225%

For these groups of numbers:

a 7

Are these statements true or false? a

4.6% < 170%

Write these quantities as percentages: a

12 out of 100

15 out of 24

22 out of 33

39 out of 60

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Find: a

33 as a percentage of 150

27 as a percentage of 75

12 as a percentage of 20

54 as a percentage of 100

18 L as a percentage of 45 L

Find: a

350 g as a percentage of 1 kg

15 cm as a percentage of 60 cm

$12 as a percentage of $80

5.5 hrs as a percentage of 8 hrs

80 pages as a percentage of 320 pages

A car weighs 1050 kg and a truck weight 3 tonnes. Express the weight of the car as a percentage of the weight of truck.

A box of fruit contains 7 apples, 7 mandarins and 6 pineapples. a

Find the total number of pieces of fruit inside the box.

What percentage of the box are apples?

What percentage of the box are mandarins?

What percentage of the box are pineapples?

In a survey of 500 people, 175 people prefer tea over coffee. What percentage of people prefer tea?

Out of 200 apples, 15 apples are rotten. What percentage of apples are rotten?

Write the fraction or number that completes these statements: a 5% of a quantity is the same as ⬚ of that quantity. b 200% of a quantity is the same as ⬚ of that quantity.

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Determine the value without using a calculator: a 25% of 16 b 500% of 11 c e 45% of 160 f 200% of 13 g

350% of 28 150% of 18

d h

60% of 30 80% of 40

Which is greater? a

30% of 30 or 250% of 14

70% of 30 or 450% of 16

13% of 17 or 29% of 13

36% of 48 or 24% of 63

Calculate 25% of 80.

Calculate 80% of 25.

What do you notice about 25% of 80 and 80% of 25?

Does this result apply in general? Investigate by writing expressions for calculating r% of s and s% of r.

A theatre has a capacity of 3000 people. On the opening night the theatre was 79% full. On the second night it was 76% full. a

How many people attended the theatre on the first night?

How many people attended the theatre on the second night?

How many people attended the theatre in total during the first 2 nights.

Mathspace WA Curriculum – Year 7 mathspace.co

Neil is eating a box of cookies. The nutritional information states that each cookie contains 5% of his recommended daily intake of sugar. If his recommended daily intake of sugar is 37 500 mg, how many milligrams of sugar does each cookie contain?

Hiraki’s salary is $85 000 per year. She is given a pay rise of 5%. a

What is the amount of the pay rise?

What would her new annual salary be?

How much would she be paid per week at the new salary level? Give your answer to two decimal places.

What percentage rise would she have needed to give her a salary of $90 000? Give your answer to two decimal places.

Use a calculator to convert these to decimals: a e

b 70.2% × 93.4

10.25%

c g

42% × 88%

d h

72% ÷ 3

Use a calculator to determine the value of x: a

90% of x is equal to 38.7

70% of x is equal to 30.1

60.1% of x is equal to 24.04

30% of x is equal to 15.9

Did you know?

Next time you’re shopping and see a sale, you can quickly calculate your savings using percentages. If a $100 jacket is 25% off, multiply 100 by 0.25 to find the discount—$25! Now you know the jacket costs $75. Percentages are everywhere, especially during sales and they help us make quick decisions on whether to snag a great deal or keep browsing. Whether it’s clothes, electronics or groceries, understanding percentages can save you money!

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Big ideas The concept of ratios and their applications allows for effective analysis and comparison of relationships between quantities, helping to solve real-world problems and making informed decisions.

Curriculum Outcomes Explore and explain the use of ratios and fractions to compare numbers and quantities. Make connections between equivalent fractions and between equivalent ratios

7 Ratios Chapter outline 7.01 7.02 7.03

Identify ratios Equivalent and simplified ratios Find part or whole measures Chapter 7 review

270 275 282 290

Architects use ratios to create scale models, which help them visualise the final structure before it’s built!

7.01 Identify ratios After this lesson, you will be able to… • use ratios to express relationships between two quantities. • describe part-to-part and part-to-whole relationships.

Ratios A ratio compares the relationship between two (or more) values. It tells us how much there is of one thing compared to another. If we want to describe the relationship between the number of blue dots and the number of green dots, we could say that there is 1 blue dot for every 3 green dots. We could also express this as a ratio, which we would write as 1 : 3. It is important that both quantities are in the same units; otherwise, we cannot compare them with a ratio. In the example above, we are comparing one blue dot to three green dots, so a single dot can be thought of as our unit. A ratio can express a “part to part” relationship, like we saw in the dot example above. We can also describe a “part to whole relationship”. For example, if we wanted to describe the ratio of green dots to all the dots, we could write it as 3 : 4, because there are 3 green dots and 4 dots in total. The order that the words are written corresponds to the order of the values in the ratio, so it is important that we don’t jumble them around. We can also express fractions, decimals and percentages as ratios.

Example 1 Write a numerical ratio for the number of circles to squares. Give your answer in the form a : b.

Create a strategy As the unit here is shapes, we can compare circles to squares directly. Count the number of circles and squares. Write the ratio as the number of circles to squares.

Apply the idea There are 5 circles and 14 squares. We can write the ratio as 5 : 14.

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Reflect and check Note that the order is important. In this question, we were asked for the ratio of circles to squares. If we were asked for the ratio of squares to circles, the answer would be 14 : 5 instead.

Idea summary A ratio compares the relationship between two values. We write ratios in the form a : b which is read as “a to b”.

Convert between units Often we might want to compare two quantities that have different units, such as a number of minutes compared to a number of hours, a distance in kilometres to a distance in metres, a duration in days to a duration in weeks and so on. To compare these types of quantities, we will need to convert one of the quantities to use the same units as the other. It does not matter which one we convert, we will end up with exactly the same ratio in the end. The final simplified ratio should not have units, it should just appear as 7 : 15, for example. We cannot remove the units until all parts of the ratio are in the same units.

Example 2 Write 31 minutes to 2 hours as a ratio.

Create a strategy Convert the hours to minutes and express the similar quantities as a ratio.

Apply the idea 1 hour = 60 minutes 2 hours = 2 × 60 minutes = 120 minutes

Multiply the hours by 60 Evaluate the product

We can write the ratio of 31 minutes to 120 minutes as 31 : 120.

Idea summary To compare quantities with different units, we need to convert one of the quantities to the same units as the other. Then we can write it in the form a : b without units.

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7.01 Practice questions Understanding 1

4:3

3:4

5:7

What does the ﬁrst number in the ratio represent?

What does the second number in the ratio represent?

Express the ratio of cats to dogs in the form a : b. 3, 4a, 5, 6aceg, 7a-d, 8a, 9a-d, 10

3, 4a, 5, 6aceg, 7c-f, 8a, 9b-e

The number of circles to squares:

6:9

3, 4b, 5, 6bdfh, 7e-h, 8b, 9c-f

The number of squares to circles:

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Write a numerical ratio for: a

Trailblazer 1cd, 2

Given a ratio of 3 cats to 5 dogs:

Fluency Ex 1

Adventurer 1bc, 2

A ratio can be described as a word phrase. E.g. the ratio 1 : 2 could represent 1 pencil and 2 pens. Write your own word phrase to represent each given ratio: a

Explorer 1ab, 2

For each diagram: i

Write the ratio of shaded squares to non-shaded squares.

Write the ratio of shaded squares to total squares.

Mathspace WA Curriculum – Year 7 mathspace.co

For each diagram: i

Write the ratio of shaded sectors to non-shaded sectors.

Write the ratio of shaded sectors to total sectors.

Ex 2

Express each pair of quantities as a ratio in the form a : b a

$5 to $7

47 volts to 43 volts

13 minutes to 17 minutes

37 lollies to 17 lollies

5 seconds to 26 seconds

13 newtons to 29 newtons

15 oranges to 76 oranges

130 days to 171 days

Express each pair of quantities as a ratio using the same units: a

9000 metres to 8 kilometres

$1 to 31c

2 centuries to 409 years

21 minutes to 2 hours

6 minutes to 659 seconds

43 weeks to 1 year

37 months to 2 years

13 hours to 5 days

Write the ratio of triangles to squares.

For each diagram: i

Write the ratio of circles to squares.

Write each statement as a ratio: a

There are 11 cats to 6 dogs.

In a classroom, there are 12 girls to 5 boys.

When making pasta, you need 5 cups of plain ﬂour to 8 eggs.

There are 3 red marbles to 2 blue marbles.

The Great Dane weighed twice as much as the Labrador.

In a working day, a bricklayer lays 50 bricks every hour.

In a bag there are 20 marbles. If there are 5 green marbles, 6 blue marbles, 2 red marbles and 7 yellow marbles, find the ratios of the coloured marbles: a

Green to red

Blue to red

Green to blue

Yellow to the total number of marbles

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Reasoning

11, 13

11–13

A fruit salad contains 4 apples, 6 oranges and 8 strawberries. Explain what the ratio 6 : 18 would represent.

A rectangular pool has a length of 12 metres and a width of 6 metres. Lucidia states that the ratio of the length to the width of the pool is 12 : 6. Annalee states that the ratio of the width to the length of the pool is 6 : 12. Determine who is correct and explain your reasoning.

A bridge is made up of 3 lengths: PQ (length 1 m), QR (length 4 m) and RS (length 7 m) as shown in the ﬁgure: 1m P

4m Q

7m R

Find the ratio of PR to the total length of the bridge and write a sentence explaining the process you used to determine the ratio.

Problem-solving 14, 15, 16a

14, 15, 16ab, 17ab

15–17

A car travels 90 kilometres in 3 hours and continues for another 150 kilometres in 5 hours. Write the distance travelled in the ﬁrst 3 hours to the total distance travelled as a ratio.

Sophia wants to make a perfume using essential oils. The recipe requires a ratio of 1 : 4 of lavender essential oil to carrier oil.

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If Sophia has 20 ml of lavender essential oil, how many litres of carrier oil does she need?

If Sophia has 100 ml of carrier oil, how many litres of lavender essential oil does she need?

In a triathlon, the swimming leg takes 20 minutes, the cycling leg takes 1 hour and 20 minutes and the running leg takes 3 hours and 45 minutes. a

Express the time for swimming to cycling as a simplified ratio.

Express the time for swimming to running as a simplified ratio.

Write the simplified ratio that represents running: cycling: swimming

The ratio of length to width for the sides of a rectangle is 3 : 1. a

Write the ratio of the area of the rectangle to the width.

If the width is doubled and we maintain the ratio of length to width, what happens to the measure of the length?

Write the ratio of the area of this larger rectangle to the width. Is this the same or different to part (a)? Explain why.

Mathspace WA Curriculum – Year 7 mathspace.co

7.02 Equivalent and simplified ratios After this lesson, you will be able to… • identify equivalent ratios. • simplify ratios to their most basic form. • understand the relationship between equivalent and simplified ratios.

Equivalent ratios Consider a cake recipe that uses 1 cup of milk and 4 cups of flour. The ratio of milk to flour is 1:4 What if we want to make two cakes? We would need to double the amount of milk and flour we use. Now the ratio of milk to flour is 2:8 We say that 1 : 4 and 2 : 8 are equivalent ratios. 2:8

1×2:4×2

2 sets of 1 : 4

Two ratios are equivalent ratios if both numbers in one of the ratios can be multiplied or divided by the same number to make the second ratio. 1

When the ratio 1 : 2.2 is multiplied by 10, we get the equivalent ratio 10 : 22.

2.2

× 10

10 :

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Example 1 Complete the table of equivalent ratios and use it to answer the following questions: a

Dogs 9 18 27 45

to : : : : :

Cats 5 10

Create a strategy To complete the table, we need to find equivalent ratios to 9 : 5. Equivalent ratios will be multiples of each other so we need to find what multiple to use to turn 9 : 5 into 27 : ⬚.

Apply the idea ×?

×3

×?

×5

×?

× 10

9: 5 27 : 9: 5 27 : 15

9: 5 45 : 9: 5 45 : 25

9: 5 : 50 9: 5 90 : 50

×?

9 × 3 = 27 so multiply all parts of the ratio by 3

×3

So 9 : 5 is equivalent to 27 : 15

×?

9 × 5 = 45 so multiply all parts of the ratio by 5

×5

So 9 : 5 is equivalent to 45 : 25

×?

5 × 10 = 50 so multiply all parts of the ratio by 10

× 10

So 9 : 5 is equivalent to 90 : 50

The complete table of equivalent ratios: Dogs 9 18 27 45 90

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to : : : : :

Cats 5 10 15 25 50

b If there are 270 dogs, how many cats are there expected to be? A

150

270

266

Create a strategy Find the multiple of 9 that results in 270. Multiply 5 by this number to find the number of cats.

Apply the idea We can ﬁnd the multiple by which the number of dogs has increased by dividing 270 by 9. We get a multiple of 30 and multiply to the number of cats, which is 30 × 5 = 150. So, if there are 270 dogs, there are 150 cats. The correct answer is option A.

Example 2 The number of students to teachers in a charity race is in the ratio 10 : 3. If 70 students take part in the race, how many teachers are there?

Create a strategy Determine the multiple of 10 that results in 70. Increase 3 by the same multiple to find the number of teachers.

Apply the idea We divide 70 by 10 to get the multiple. The multiple is 7. 10 : 3 = 70 : 3 × 7 = 70 : 21

Multiply 3 by 7 Evaluate

So, if there are 70 students taking part in the race, there will be 21 teachers.

Idea summary Two ratios are equivalent ratios if both numbers in one of the ratios can be multiplied or divided by the same number to make the second ratio.

Simplified ratios Ratios can be simplified in the same way we simplify fractions: • If both sides of the ratio have a common factor, we can divide both sides of the ratio by that common factor to simplify the ratio. A ratio is in its simplest form when: • We can no longer find a common factor. • The ratio contains only integers (whole numbers). 7.02 Equivalent and simplified ratios mathspace.co

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Example 3 Simplify the ratio 35 : 5

Create a strategy

Apply the idea

Find the highest common factor of the two numbers and then divide both numbers by it.

The HCF of 35 and 5 is 5, as 5 is the highest number that can divide both 35 and 5 with no remainders. Divide both numbers by the HCF. 35 ÷ 5 = 7 5÷5=1 So the simpliﬁed ratio is 7 : 1.

Example 4 a Simplify the ratio

Create a strategy

Apply the idea

Find a number that we can multiply to both

Multiply both parts of the ratio by the denominator of the fraction, which is 4.

and 2 to get a ratio involving only whole numbers.

So the simpliﬁed ratio is 3 : 8. b Simplify the ratio

Create a strategy Find a number that we can multiply to both

and

to get a ratio involving only whole numbers.

Apply the idea Get a common denominator with the fractions, which is 15.

So the simpliﬁed ratio is 9 : 2.

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Reflect and check Once we have a common denominator, we can just look at the numerators because multiplying both sides by 15, we can remove the denominators anyway.

Example 5 Write 540 cents to $3.00 as a fully simpliﬁed ratio.

Create a strategy

Apply the idea

Convert the dollars into cents, then simplify the ratio.

$1.00 = 100 cents Convert dollars to cents

Divide by 10

Evaluate

Divide by 6

Simplify

Idea summary A ratio is a simpliﬁed ratio if • •

both numbers are integers we can no longer find a common factor

7.02 Practice questions Understanding Ex 1a

1, 2

Adventurer 1aceg, 2

Trailblazer 1bdfh, 2

Complete each pair of equivalent ratios: a e

Explorer

2:1=4:⬚

2 : ⬚ = 4 : 14

b f

2 : 5 = ⬚ : 10

6 : ⬚ = 18 : 12

c g

3:9=6:⬚

⬚ : 20 = 1 : 4

d h

8:4=⬚:6

⬚ : 16 = 8 : 32

If 3 : 4 is equivalent to 12 : 16, what is the missing number in the ratio 1 : 2 = 3 : ⬚?

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Fluency

5a-e, 6–8, 9a

3–4, 5e-il, 6, 7, 9

3, 4, 5h-l, 6, 7, 9

Ex 1b

Tricia and Luigi contribute money for a birthday present in the ratio 4 : 9. If Tricia contributed $24, how much has Luigi contributed?

Ex 2

The ratio of students to teachers competing in a charity race is 9 : 4. If 54 students take part in the race, how many teachers are there?

Ex 3

Write as simplified ratios: a

Ex 5

10 : 20

20 : 40

35 : 7

100 : 2000

8:4

15 : 40

36 : 48

16 : 24

45 : 35

18 : 22

450 : 350

10 : 100

Write down three ratios that are equivalent to these ratios: a

Ex 4

4 : 10

3:1

6 : 10

2:5

Write as simplified ratios: a

7 : 6.3

1.5 : 2.4

0.5 : 2.25

2.4 : 3.9

2.8 : 1.6

0.4 : 2.08

$0.60 : $2.00

$5.20 : $9.50

$8.20 to $11.60

5.4 km to 420 m

120 cm to 5.1 m

11 years to 28 months

20 days to 5 weeks

Write as simplified ratios: a

Reasoning

10, 11, 13ab, 14a

12, 13a-c, 14a

12, 13d, 14b

At a sporting event, the ratio of security guards to spectators is 1 : 120. If 8400 spectators attend the event, explain how to ﬁnd how many security guards will be required.

Suppose you are explaining to a classmate who missed the lesson how to simplify the ratio 620c to $4.00. Describe how you would simplify the ratio.

Given the ratio of apples to oranges is 3 : 5, explain whether these statements are true or false.

280

If the number of apples is increased, the number of oranges will also increase.

If the number of apples is doubled, the number of oranges is halved.

If the number of apples is doubled, the number of oranges is doubled.

If there are no apples, there are no oranges.

Mathspace WA Curriculum – Year 7 mathspace.co

Hermione is making a patterned lid for a wooden box. She knows that for every mahogany, she will need 70 centimetres of oak. a

She states the ratio of mahogany to oak is why not.

Write the ratio of mahogany to oak using fractions.

Simplify this ratio.

How many metres of mahogany would Hermione need if she wants to use 14 metres of oak?

Is Hermione correct? Explain why or

In these sets of equivalent ratios, one ratio is incorrect. Find the incorrect ratio in each set. Explain why you think it doesn’t ﬁt in the set and replace one of its numbers to make it equivalent to the others. a

18 : 24,

3 : 4,

6 : 9,

15 : 20,

30 : 40

5 : 10,

20 : 40,

14 : 30, 2 : 4,

8 : 16

Problem-solving 15ab, 16ab 15

metres of

15–17

A painter wants to create a certain colour by mixing two different colours of paint, Vespa and Nitro, in the ratio 4 : 1. He uses 4 litres of the Nitro colour. a

How many litres of the Vespa colour must he use?

How many litres of paint will he have altogether once the two colours are combined?

A litre of paint can cover an area of 8 m2. Suppose that the rooms in a house each have 65 m2 of wall space. How many rooms can the painter paint with the new colour?

The ratio of students who play badminton to football to softball is 2 : 5 : 9. a

If there are 72 students who play softball, how many students play badminton?

Using your answer from part (a), determine the total number of students that play these sports.

A local business wants to sponsor the badminton and football teams by buying their team shirts. If shirts are $15 each, can the business provide enough shirts for the teams with a budget of $800?

Two kinds of pine trees, Bristlecone and Aleppo, are planted in rows. In each row, the ratio of Bristlecone to Aleppo is 10 : 9. There are a total of 1140 trees planted. Determine the number of each Bristlecone and Aleppo that were planted.

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7.03 Find part or whole measures After this lesson, you will be able to… • apply your knowledge of equivalent and simplified ratios to calculate shares of a quantity and determine a quantity from a given share.

Find a share of a quantity Suppose that Opie and Josie contribute some funds to a business in the ratio of 7 : 4. The total amount invested is $88 000. We consider the $88 000 as the total 11 parts. This means that Opie contributed 7 parts and Josie contributed 4 parts. There is $88 000 in total, which needs to be equally shared into 11 parts. $88 000

7 parts

4 parts

Find the size of each part: $88 000 ÷ 11 = $8000. $88 000

$8000 $8000 $8000 $8000 $8000 $8000 $8000 $8000 $8000 $8000 $8000

7 parts

• Opie invested the value of 7 parts: $8000 × 7 = $56 000 • Josie invested the value of 4 parts: $8000 × 4 = $32 000

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4 parts

Example 1 Ben always buys melons and bananas in the ratio 7 : 3. If he buys 30 pieces of fruit in total, how many melons did he buy?

Create a strategy Divide the amount of fruit into equal parts. The ﬁrst part of the ratio refers to melons, so 7 parts will be melons.

Apply the idea The ratio 7 : 3 means we have 7 + 3 = 10 parts. 30 pieces of fruit

7 parts

3 parts

7 of those parts are melons. 30 pieces of fruit M

7 parts

30 ÷ 10 = 3 7 × 3 = 21

3 parts

Divide 30 into 10 parts 7 parts are melons

Ben buys 21 melons (and 9 bananas).

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Example 2 In a zoo, the ratio of elephants to lions is 8 : 3. If there are 66 elephants and lions altogether, how many lions are there?

Create a strategy Divide the number of animals into equal parts. The second part of the ratio refers to lions, so 3 parts will be lions.

Apply the idea The ratio 8 : 3 means we have 8 + 3 = 11 parts. 3 of those parts are lions. 66 animals

8 parts

66 ÷ 11 = 6

3 parts

Divide 66 into 11 parts 66 animals

E E E E E E E E E E E E E E E E E E E E E E E E L L L E E E E E E E E E E E E E E E E E E E E E E E E L L L 8 parts

3 × 6 = 18

L L L L L L

3 parts

3 parts are lions

There are 18 lions (and 48 elephants).

Idea summary When we have a ratio such as 2 : 3, we have a total of 5 equal parts. We can divide a quantity by the number of equal parts to ﬁnd one part. Then we can give the correct number of these to each part of the ratio.

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Find a quantity from a share A recipe uses oats and flour in the ratio of 3 : 4. These are known as the dry ingredients. If we know we are to use 180 g of oats, how much flour should we use? And what is the total amount of the dry ingredients? In this case, we know the amount of oats and we want to work out the amount of flour. The 3 in the ratio 3 : 4 refers to the oats and this is 180 g. 180 g

3 parts

4 parts

Oats

Flour 3

One part would therefore be 180 ÷ 3 = 60 g. Flour is then 4 parts, so 4 × 60 = 240 g of ﬂour should be used. 180 g

60 g

3 parts

60 g

Oats

4 parts

60 g

3 : 4 180g :

Flour

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The total of the dry ingredients (oats and ﬂour) is then 3 + 4 = 7 parts, so the total of the dry ingredients is 7 × 60 = 420 g. 420 g

180 g

60 g

240 g

60 g

3 parts 60 g

60 g

4 parts 60 g

Oats

60 g

3 : 4 180 g : 240 g

60 g

Flour

Example 3 Juliet wants to make some cakes. She has lots of ﬂour, but only 2 kg of butter. The ratio of butter to flour in the recipe is 4 : 5. How much ﬂour should she use?

Create a strategy Create a ratio diagram that shows the ratio of butter to flour (4 : 5).

Apply the idea We need to divide the 2 kg of butter into 4 parts. 2 kg

4 parts

2 ÷ 4 = 0.5

5 parts

2 kg divided into 4 parts 2 kg

0.5 kg

4 parts

5 × 0.5 = 2.5

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0.5 kg

5 parts

Each part is worth 0.5kg. Multiply the 5 parts of the flour by 0.5.

Mathspace WA Curriculum – Year 7 mathspace.co

4.5 kg

2 kg

0.5 kg

2.5 kg

0.5 kg

4 parts

0.5 kg

5 parts

Juliet should use 2.5 kg of ﬂour.

Idea summary When we have a ratio such as 2 : 3, we have a total of 5 equal parts. We can use the number of parts of the ratio to determine the size of one part. Once we know the size of one part, we can use this to determine the size of each side of the ratio or the total amount.

7.03 Practice questions Understanding

1, 2

Adventurer 1–3

Trailblazer 2, 3

The ratio of lollies shared between Harris and Bella is 5 : 4. How do you determine the total number of parts of lollies that they have?

James and Emma scored goals in their netball game in the ratio 4 : 3. Write the ratio of James’ goals to Emma’s goals to the total scored goals.

56 building blocks are shared between Mohamad and Isabelle in the ratio 2 : 5. How do you determine the number of blocks that Mohamad and Isabelle each have?

Fluency Ex 1

Explorer

4–7

5, 7, 8

Hiro always buys melons and bananas in the ratio 7 : 3. If he buys 50 pieces of fruit in total, how many melons did he buy?

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Ex 2

The ratio of adults to children on a train is 2 : 7. The train is carrying 360 passengers. a

Ex 3

How much money does James get?

Simplify the ratio.

How much did Xavier pay for the groceries?

How much money does Sophia get?

A car and a truck are driving down the highway. Their speeds are in the ratio 3.1 : 2.9. If the car is travelling at 93 km/hr, how fast is the truck travelling? 11, 12

11, 12

9–12

Mara is mixing paint for a mural. She uses two colours in the ratio of 3 parts of blue paint for every 6 parts of yellow paint. a

Write the simplified ratio of blue paint to yellow paint.

Describe how to determine the quantity of yellow paint needed if Mara uses 9 parts of blue paint.

Avril is making a scarf that uses two materials. She wants to use the materials in the ratio of metres of red for every

How many children are on the train?

Xavier and Ursula split the cost of the groceries in the ratio 0.5 : 1.5. Altogether, the groceries cost $160.

Reasoning 9

James and Sophia share $880 in the ratio 2 : 9. a

How many adults are on the train?

metres of green.

Write the simpliﬁed ratio of red material to green material.

If she needs 33 metres of red material, describe how to ﬁnd how many metres of green material she will need.

Kathleen and Luke invest money in a business in the ratio 3 : 10. A student reading the statement above concludes that if Kathleen invests $3000 in the business, that means Luke invested $7000 for a total of $10 000 invested.

What is the mistake the student may be making with their statement.

Rewrite the statement so that the student’s conclusion is correct.

Two kinds of pine trees, Paperbark and Gumtree, are planted in rows. In each row, the ratio of Paperbark to Gumtree is 9 : 8. Altogether, 340 trees are planted. Explain two different ways to determine how many Gumtree trees are planted.

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Problem-solving 13, 16 13

13, 15, 16

13–16

The perimeter of a rectangle is 42 cm and the ratio of its length to its width is 4 : 3. a

How many parts are in the ratio?

Find the width of the rectangle.

Find the length of the rectangle.

Neil is mixing paint to get a speciﬁc shade of orange. He wants a mix of red and yellow paint in the ratio of . How many litres of red paint will Neil need if he wants to make 75 litres of orange paint?

The ratio of red to green to black coloured area on a dart board is 12 : 7 : 31. The area of the dart board is 150 cm2. a

Find the size of a single ratio part.

Find the area corresponding to each colour:

Red area

Green area

iii

Black area

Vanessa competed in a triathlon and found that her times for the swimming, cycling and running sections were in the ratio 3 : 1 : 2. If the quickest section took 720 seconds, how many minutes did she take to complete the whole triathlon?

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7 Chapter review 1

Express each pair of quantities as a ratio: a

60 : 40

90 hours to 150 hours

45 : 15

525 : 375

Write as simplified ratios: a

18 apples to 54 apples

Write as simplified ratios: a

24 pounds to 42 pounds

0.75 : 1.5

3.6 : 2.7

1.2 : 3.3

16 years to 48 months

30 days to 6 weeks

Write as simplified ratios: a

7.2 km to 600 m

Emma and Marco contribute money to an investment in the ratio 3 : 7. If Emma has invested $3000, how much has Marco invested?

The ratio of players to coaches participating in a community sports event is 7 : 3. If 49 players take part in the event, how many coaches are there?

A train travels 120 kilometres in 4 hours and continues for another 180 kilometres in 6 hours. Write the distance travelled in the first 4 hours to the total distance travelled as a ratio.

An artist wants to create a unique shade by blending two different colours of paint, azure and tangerine, in the ratio 3 : 2. She uses 6 litres of the tangerine colour.

How many litres of the azure colour must she use?

How many litres of paint will she have altogether once the two colours are combined?

A litre of paint can cover about 10 m2. Assume that the walls in a studio each have 50 m2 of surface area. How many studios can the artist paint with the new shade?

The ratio of students who play basketball to soccer to volleyball is 3 : 4 : 7. a

If there are 84 students who play volleyball, how many students play basketball?

Using your answer from part (a), find the total number of students that play soccer and volleyball.

A local company wants to sponsor the basketball and soccer teams by buying their team jerseys. If jerseys are $20 each, can the company provide enough jerseys for the teams with a budget of $1000?

Two varieties of apple trees, Gala and Honeycrisp, are planted in an orchard. In each row, the ratio of Gala to Honeycrisp trees is 5 : 6. There are a total of 1210 trees planted. Find the number of each type of tree that was planted.

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Every week, Olivia buys oranges and pears in the ratio 3 : 4. If she buys 42 pieces of fruit in total, how many oranges did she buy?

Sophie and Ethan split the cost of a meal in the ratio 0.7 : 1.3. Altogether, the meal costs $120. How much did Sophie pay for the meal?

A bicycle and a scooter are moving down the street. Their speeds are in the ratio 2.4 : 3.6. If the bicycle is travelling at 18 km/hr, how fast is the scooter travelling?

The perimeter of a parallelogram is 54 cm and the ratio of its adjacent sides is 5 : 4.

Find the length of the longer side of the parallelogram.

Find the length of the shorter side of the parallelogram.

Adjacent sides

Lily is mixing paint to get a specific shade of purple. She wants a mix of blue and pink paint in the ratio of

How many litres of blue paint will Lily need if she wants to

make 80 litres of purple paint? 17

The ratio of the number of boys to the number of girls in a school is 2 : 3. The ratio of the number of teachers to the number of girls is 5 : 33. If there are 275 students in the school, how many teachers are there?

A recipe for a smoothie calls for fruit, yoghurt and milk to be in the ratio of

measured in fractions of a cup. How many cups of milk are needed if we use 2 cups of fruit to make the smoothie?

Did you know?

In sports, ratios help analyse player performance! For example, if a basketball player scores 30 points in 10 shots, their scoring ratio is 3:1, meaning they score 3 points for every shot taken. This information can help coaches strategise better and improve player performance. Chapter 7 review mathspace.co

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Big ideas Algebraic expressions serve as the fundamental units of algebra, enabling the representation and analysis of real-world scenarios.

Curriculum Outcomes • Use real-world contexts or concrete materials to introduce the concept of a variable to represent a number using a letter. Create simple algebraic expressions and evaluate by substituting a given value for the variable/s • Extend and apply the associative and commutative laws and properties of numbers to include variables

8 Algebra Chapter outline 8.01 Introduction to variables 8.02 Build algebraic expressions 8.03 Substitution 8.04 Collect like terms Chapter 8 review

294 299 305 311 317

Emoji are symbols used to represent different words, objects and feelings. In algebra we use letters as symbols to represent unknown numbers.

8.01 Introduction to variables After this lesson, you will be able to… • write algebraic expressions using variables. • combine like variables in algebraic expressions.

Write expressions with variables The basic concept of algebra is using symbols, called variables or pronumerals, in place of numerical values that we don’t know. We normally use letters of the alphabet as our variables. We can use algebra tiles to help us visualise algebraic expressions.

Let this equal x

How many oranges in the box?

The tile x represents an unknown number. The tile + 1 represents adding one unit and −1 represents subtracting one unit. Positive

Negative

Variable tiles

−x

Unit tiles

−1

This table demonstrates how expressions can be built using the tiles: Word expression

Algebraic expression

three more than x

x+3

three less than x

x–3

Representative with algebra tiles +x +1

+1 +x

−1

−1 +x

the product of x and three

x×3

+x +x

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−1

Example 1 Write the algebraic expression represented by these algebra tiles: +1

Create a strategy Count the number of

tiles. Then count the number of +1 tiles.

Apply the idea

Reﬂect and check

There is one + x tile and four +1 tiles. The expression is x + 4.

The expression represents four more than the unknown number x.

Idea summary We use variables to represent unknown numbers in algebra.

Combine the same variables We can combine the same variables. For example:

2×

In algebra, 2 × p is written as 2p. There is no need to write the multiplication sign.

If there are 5 groups of p, 5 × p, we can write it as: 5p Now, suppose we have one p and one q:

p q

p+q This cannot be simpliﬁed as p and q are not the same variable. 8.01 Introduction to variables mathspace.co

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Example 2 Write a simpliﬁed algebraic expression for these algebra tiles: +x

Create a strategy Count the number of

tiles. Then count the number of +1 tiles.

Apply the idea Remember that the value of five +1 tiles is 5 and the value of two

tiles is 2x.

The expression is 2x + 5.

Reﬂect and check 2x + 5 means ﬁve more than double the value of x.

Idea summary We can combine the same variables: x+x+x=3×x = 3x In algebra, there is no need to write the × sign.

8.01 Practice questions Understanding

Explorer 1–5

Adventurer 1–5

Trailblazer 1–5

What is a variable used for in algebra?

Farmer Jones has sheep, cows, goats and chickens on his farm. Use variables to write an expression for the total number of animals on the farm.

Name the variable(s) used in these algebraic expressions: a

4c + 7d

mn + 1

2g + 3

Simplify these algebraic additions by expressing each using multiplication: a

a + a + a

p+p+p+p+p+p

m + m + m + m

k+k+k+g+g+g

12 × d

Simplify these algebraic multiplications: a

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3×a

8×k

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5×4×p

Fluency 6

6, 7, 8a-c, 9a-c

a b c

Ex 1, 2

7cd, 8cd, 9cd

Consider this diagram:

6–9

Fill in the boxes to write an algebraic expression that represents the diagram: ⬚ + ⬚ + ⬚ + ⬚ + ⬚.

Fill in the box to write an algebraic expression that represents the diagram: 5 × ⬚. Write the answer to part (b) in another way.

Write a simpliﬁed algebraic expression for these diagrams: a

Each single box represents p.

Each single box represents b.

Each single box represents y.

ach purple box represents p and each E green box represents g.

Write a simpliﬁed algebraic expression for these diagrams: a

+1 +1 +1 +x

+x −1

+x +x

−x

−1

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Use these algebra tiles to draw a diagram that represents each expression:

Positive

Negative

Variable tiles

−x

Unit tiles

−1

x+2

Reasoning

2x + 1

10, 11

3x − 4

−3x + 2

12, 13

Show that a + a + a + a = 4a for three different values of a.

Show that x + x + x + y = 3x + y for three different values of x and y.

Using the given key, explain why it is not possible to simplify a + b any further using algebra.

Key

contains a apples

contains b apples

Is 2x + 3 the same as 3 + 2x? Explain your answer.

Problem-solving 14, 15, 16a 14

14, 15, 16ab

15, 16bc, 17

Consider this list of colours of pens: blue, blue, red, red, red If b is the number of blue pens and r is the number of red pens:

What does the expression b + r represent?

What are the values of b and r?

Blue marbles are represented by b, green marbles by g, red marbles by r and yellow marbles by y. b + g + r + y represents the total number of marbles in the marble jar. What are the values of b, g, r and y?

Complete the pattern in these tables: a

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1 2 3 4 x 2x 1 z+2

2 2z + 4

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1 2y + 1

2 4y + 2

Consider the key: Key contains a oranges contains b oranges

If James has 5 of the boxes containing b oranges and this totals 100 oranges, what is the value of the variable b?

If Denise has 3 of the boxes containing a oranges and this totals 33 oranges, what is the value of the variable a?

Sasha has 6a + 2b oranges. How many oranges is this?

8.02 Build algebraic expressions After this lesson, you will be able to… • recognise and identify coefficients, variables, and constant terms in algebraic expressions. • translate word expressions into algebraic expressions and simplify them when necessary.

Terms, variables and coefficients We use algebraic expressions when we want to write a number sentence but don’t know all the numbers involved. For example: What is the total weight of a cat and a dog? In this case, let’s use c for the weight of the cat and d for the weight of the dog.

c+d

c + d is called an algebraic expression. c and d are called variables or pronumerals. They are used in place of a numeric value. Coeﬃcients are used to show how many lots of a variable we Total weight = cat plus dog = c + d have. The variable u with a coeﬃcient of 3 is written as 3u, which means 3 × u.

3u = 3 × u The variable is u with a coefficient of 3

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Consider the expression:

• This is an expression with 5 terms. • The term 4x has a coefficient of 4. • The term − 3y has a coefficient of − 3. The negative belongs with the coefficient. • The term

has a coeﬃcient of

• k is equal to 1 × k, which is also equal to 1k, so it has a coeﬃcient of 1. • The term 7 has no variable. It is called a constant term.

Example 1 For the expression

, identify:

a the coefficient of y

Apply the idea

coefficient of y

b the constant term

Apply the idea

constant term

Idea summary In algebra, letters, called variables or pronumerals, are used to represent unknown numbers. A term consists of a number and a variable. For example: 2x, −3y and

are terms.

A coefficient is a number that is placed before the variable in an algebraic term. For example: −3 is the coefficient of −3y. If there is no number placed before the variable then the coefficient is 1. For example: w has a coefficient of 1. A constant term is a term with no variable. For example: 2, −5 and 32 are constant terms. An algebraic expression is a combination of numbers and variables with mathematical operators. For example: 2x − 5y + 11 is an expression.

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Basic operations in algebra The basic operations work almost the same for algebraic terms as they do for numbers. Between variables and numbers, we have: Word expression Three more than x Three less than x

Algebraic expression x+3 x−3

The quotient of x and three

x÷3

The product of x and three

x×3

Simpliﬁed algebraic expression x+3 x−3

Between variables and other variables, we have: Word expression y more than x y less than x

Algebraic expression x+y x−y

The quotient of x and y

x÷y

The product of x and y

x×y

Simpliﬁed algebraic expression x+y x−y

Notice that division is written as a fraction, for example:

Example 2 What does the expression 8x mean?

Create a strategy

Apply the idea

A coeﬃcient is multiplied by the variable next to it.

The expression 8x means 8 is multiplied by x.

Example 3 Write an algebraic expression for the following phrase, “eight more than the quotient of 9 and x”.

Create a strategy

Apply the idea

Translate the terms into mathematical symbols and operations.

The phrase “eight more than” indicates that we need to add 8. The “quotient of 9 and x” indicates division, which we can write as a fraction, with 9 as the numerator and x as the denominator. We can combine the whole description into a single expression:

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Idea summary In algebra, we do not need to write multiplication signs; 4 × x is written as 4x. In algebra, we write division as a fraction; y ÷ 4 is written as .

8.02 Practice questions Understanding

Variable

12 in the term 12x

Coefficient

b in the term 5b

iii

Expression

Constant

2x + 5y

Number of terms

Coeﬃcient of x

Coeﬃcient of y

Constant term

What is the coeﬃcient in these terms? a

−2y

−6y

0.5m

8ab

−101pqr

−i

πr

5x + 8

What is the constant term in these expressions? a

u + 11

8y − 3

14 − 20ab + 4a

a − 11

10a − 12b + 7c − 2 h

15v + 8y + 2uv − 30

Write down the number of terms in these expressions: a

−2x

7y + 1

1024

ab + 18

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Trailblazer 1, 2, 3eh-l, 4e-j, 5bdfh, 6

For the expression −15x + 2y − 3, write down the:

i 4

Adventurer 1, 2, 3e-j, 4c-h, 5aceg

Match the term on the right with the appropriate example from the left.

a 3

Explorer 1, 2, 3a-dij, 4a-f, 5aceg

8xy + 4 − 12y

Mathspace WA Curriculum – Year 7 mathspace.co

5x − 7y + 8xyz − 12

3xy + 5 + 4y − x

Choose the correct definition of these terms that matches the options: i

Sum

The result of multiplying two or more values or algebraic expressions.

Product

The result of dividing a value or algebraic expression by another.

iii

Difference

A single component of a product.

Quotient

The result of adding two or more values or algebraic expressions.

Factor

A single component of a sum or subtraction.

Term

The result of subtracting a value or algebraic expression from another.

7ab, 8, 9ab, 10a-c, 11a, 13–16

Fluency Ex 1

the coefficient of x

the constant term

8y + 2x − 3y + 8

5x − 2y + 4

13x − 13y + 13z − 13

x−6

e 8

Ex 3

p+7

Five times the sum of x and ten

Three lots of the sum of a and b

Seven lots of the difference between n and 8, then increased by 4

Eleven more than the product of 4 and the difference between m and 9

Write an algebraic expression for these statements without using a multiplication or division sign: Seven multiplied by x

Eight more than 5x

Three less than 8y

The product of 9a and b

The quotient of 8u and 9

Four times y take away 9

7 less than the quotient of 5 and x

The sum of a and b, divided by 6

4(n + 1) − 11

Write a word statement for these expressions: a

Write an algebraic expression for these statements without using a multiplication or division sign:

−x + y − z

Write a word statement for these expressions: a

7b-e, 8, 9cd, 10c-h, 11b-d, 12bdfh, 15, 16

For each expression, identify:

Ex 2

7b-d, 8, 9a-c, 10ac, 11a-c, 12aceg, 13, 15, 16

2(b + 9)

x (3 − y)

For these expressions: i

Identify the order in which the operations should be performed.

Simplify the expression.

3×u÷v+8

5+y×x−4

(4 − k) ÷ 5

7 − 12 ÷ (3 × a)

8+3×3÷m

(n + 3 − 2) ÷ 7

(15 − 3) × (m + 1)

(3 × p + 5 − 4) ÷ 6

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The cost of a movie ticket is $12. Write an algebraic expression for the cost of n movie tickets.

The total cost of buying x apples is $4. Write an algebraic expression for the cost of one apple.

A shipping company charges a ﬂat fee of $25 plus $0.50 per kilogram for shipping packages. Write an algebraic expression for the total cost of shipping a package that weighs w kilograms.

The length of a rectangle is 4 more than its width. Write an algebraic expression for the length of the rectangle in terms of its width, w.

Reasoning

17, 18

17–19

17–20

Why does 5 × 7 + 2 need to have a multiplication sign, however 3b + 2 doesn’t?

Write “the product of 3 and k, then divided by b” in three different ways.

If m is a positive odd number:

Explain why m + 2 and m + 4 are also odd.

Write the third odd number after m.

Write the sixth odd number after m.

Write the pth odd number after m.

Explain how “2 groups of 8 more than x” is different to “8 more than 2x.”

Problem-solving 21

21, 22

22–25

A recipe for a cake calls for 2 cups of ﬂour for every 3 eggs. Write an algebraic expression for the number of cups of ﬂour needed for n eggs.

The sum of the ages of a mother and her daughter is 42 years. If the mother’s age is q years, write an algebraic expression for the daughter’s age.

Write down an algebraic expression for:

The total value of m coins, where each coin is worth p cents.

The length of string remaining when two pieces of length K cm are cut from a piece of string originally L cm long.

The term “consecutive integers” means whole numbers that follow each other on the number line. For example: 5, 6, 7 are three consecutive integers. a

Write algebraic expressions for three consecutive integers if the smallest integer is x.

Write algebraic expressions for three consecutive integers if the largest integer is p.

Write algebraic expressions for three consecutive integers if the middle integer is y.

A square has an area of h2 square units. Write an algebraic expression for the perimeter of the square.

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8.03 Substitution After this lesson, you will be able to… • use substitution to evaluate algebraic expressions. • complete tables of values and find unknown values in a table.

Evaluate expressions To evaluate an algebraic expression, we ﬁrst replace the variable with a number. We call this substitution. If p = 2 then: • p + 4 is equal to 2 + 4 which is 6. • 10 − p is equal to 10 − 2 which is 8. • 3p is equal to 3 × 2 which is 6. Each time we replace the p with the number 2 and then evaluate.

Example 1 Find the value of 9 + m when m = 3.

Create a strategy We can substitute m = 3 into the equation by replacing m with 3.

Apply the idea 9+m=9+3 = 12

Substitute m = 3 into the equation Evaluate the addition

Example 2 Find the value of

when u = 54.

Create a strategy We can substitute u = 54 into the equation by replacing u with 54.

Apply the idea Substitute u = 54 into the equation

Evaluate the division

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Example 3 Evaluate x2 + x when x = 5.

Create a strategy We can substitute the value of 5 into all the places we see an x; remember that x2 means x × x.

Apply the idea x2 + x = 52 + 5

Substitute the value of x

= 25 + 5

Evaluate 52

= 30

Evaluate the addition

Example 4 Evaluate 6x + 4y + 6 when x = 3 and y = 5.

Create a strategy We can evaluate the expression by substituting in the values for x and y.

Apply the idea 6x + 4y + 6 = 6 × 3 + 4 × 5 + 6

Substitute the values of x and y

= 18 + 20 + 6

Perform multiplication

= 44

Evaluate the addition

Idea summary To evaluate an algebraic expression, we substitute number(s) for the variable(s) and simplify the answer.

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Complete tables of values Substitution can be used to complete this table of values with the rule y = 3x +1: x y

1 4

2 7

3 10

4 13

5 16

20 ?

105 ?

To ﬁnd the 20th number (x = 20): y = 3x +1

Write the equation

= 3 × 20 + 1

Substitute x = 20 into the equation

= 60 + 1

Evaluate the multiplication

= 61

Evaluate

and the 105th number (x = 105): y = 3x +1

Write the equation

= 3 × 105 + 1

Substitute x = 105 into the equation

= 315 + 1

Evaluate the multiplication

= 316

Evaluate

Example 5 Consider the table: m n

10 3

11 4

12 5

a Use the formula n = m − 7 to complete the table.

Create a strategy We can ﬁnd the n-values by substituting the m-values into the formula.

Apply the idea n=m–7

Write the formula

n = 13 – 7

Substitute m = 13

n=6

Evaluate the subtraction

n = 14 – 7

Substitute m = 14

n=7

Evaluate the subtraction

n = 15 – 7

Substitute m = 15

n=8

Evaluate the subtraction m n

10 3

11 4

12 5

13 6

14 7

15 8

Reflect and check Notice the pattern that is created in the table. Each time m changes, the n value changes by the same amount.

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b What is the value of n when m = 20?

Create a strategy

Apply the idea

We can substitute m = 20 into the equation by replacing m with 20.

n=m−7

Write the formula

n = 20 − 7

Substitute m = 20

n = 13

Evaluate the subtraction

Idea summary To ﬁnd an unknown value in a table of values, substitute the known number for the variable into the rule for the table.

8.03 Practice questions Explorer

Understanding 1

Emma is x years old.

I have x pens.

The result of 10 − 2 is x.

x is one more than seven.

Substitute the number 8 in place of x:

Write new expressions by replacing the ⬚ with 3, you do not need to evaluate the expression. 5×⬚

⬚ + 17

Use the table of values to answer these questions: x

308

The apple is ⬚.

My favourite colour is ⬚.

2x + 1

b d

a 4

A ⬚ pencil.

The ﬂag is ⬚ and blue.

Trailblazer 2–5

1–5

Substitute the word “red” in place of the ⬚: a

Adventurer

1–4

1 3

3 7

5 11

7 15

1−⬚

9 19

What value of x when substituted into 2x + 1 gives a result of 3?

What value of x when substituted into 2x + 1 gives a result of 15?

What is the value of 2x + 1 if 9 is substituted for x?

Rewrite each expression with the value of the variable(s) substituted. Do not evaluate the expression. a

7p + 5 when p = 6

ab − 8 when a = 12, b = 4

when c = −1, d = 5

k − z when

Mathspace WA Curriculum – Year 7 mathspace.co

, z = −2

Fluency 6

Ex 1, 2, 3

Ex 5

m+5

−m + 8

−4

Find the value of: a

9 + m when m = 8

14 + v when v = 15

n − 2 when n = 5

n − 8 when n = 44

u × 5 when u = −6

when n = 14

when u = 54

2x + 9 when x = 4

when x = 8

3x − 4 when x = 7

n × 9 when n = 0.8

Ex 4

8m-r, 9, 10c-f

Calculate the value of x − 5, when x is equal to: a

7, 8h-l, 9ac, 10cd

If m = 6, calculate the value of: a

6, 7, 8a-g, 9ab, 10a-c

6x − 9 when x = 10

when x = 2

2(x + 7) when x = 4

+ 6 when x = 63

14x +

2a2 + 6 when a = 4

a2 + 6b2 when a = 2 and b = 3

when

Find the value of: a

m + n when m = 7 and n = 12

2x + 3y + 1 when x = 6 and y = 5

8a + 4b − 9 when a = 2 and b = 5

when x = −3 and y = 6

a3 + bc when a = 3, b = 8 and c = 7

x3 + y2 + z when x = 4, y = 8 and z = 10

Complete these tables: a

p 1 2 3 p+7 8 9 10

p 1 2 3 6p + 4 10 16 22

p 1 2 3 5p − 3 2 7 12

p 3p

x 2x + 4

p 11p + 5

1 3

2 6

3 9

1 6

2 8

3 10

1 16

2 27

3 38

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Reasoning 11

11, 12

11–13

If p = 2, calculate the value of:

2 × 12p

Explain your results from part (a).

Is this result the same as previous results? Explain.

3 × 8p

iii

6 × 4p

Evaluate 24 × (p + p) when p = 1.

1 × 24p

If x = 5 and y = 3, write three different expressions including both variables that each evaluate to 23.

If a and b are integers between 0 and 10, show that 10a + b is always a two-digit number.

Problem-solving 14, 16, 19

14, 16, 17, 20, 21

14–18, 21

When a whole number is substituted for x in the expression 5x + 1, the result is a number greater than 56. What is the smallest value that the whole number could be?

If t is an integer greater than 0 and less than 7, ﬁnd the minimum possible value of 9 − 4t.

Describe what happens to the value of the expression when:

310

the value of x increases

the value of x decreases

x + 6

15.6 − x

Describe what happens to the value of the expression when: i

the value of x doubles

the value of x halves

10x

Describe what happens to the value of the expression when: i

the value of both x and y is doubled

both x and y are multiplied by 3

2x + 3y

5xy

Valentina’s hens produce 5 eggs each day. a

If Valentina collects the eggs from her hens for y days, write an algebraic expression for the total number of eggs she collects.

Find the number of eggs Valentina will have after 30 days.

John breeds dogs for sale. Each of his d dogs gives birth to p puppies. a

Write an algebraic expression for the total number of puppies.

If he has 3 dogs, ﬁnd the total number of puppies.

In a Year 7 class, students are either twelve years old or thirteen years old. a

If there are k twelve year-old and m thirteen year-old students, write an algebraic expression for the total number of students in the class.

If there are 4 twelve year-old and 16 thirteen year-old students, ﬁnd the number of students that are in the class.

Mathspace WA Curriculum – Year 7 mathspace.co

8.04 Collect like terms After this lesson, you will be able to… • identify like terms. • combine like terms using addition and subtraction in order to simplify expressions.

Like terms Algebraic terms are called like terms if they have exactly the same variables with the same powers. It doesn’t matter what order the variables are in. 5a and 3a are like terms.

8ab and −2ba are like terms. −2b and 4c are not like terms.

p and q are not like terms

x and x are not like terms.

Example 1 Are these pairs of terms like terms: a 9y and 10y?

Apply the idea 9y and 10y are like terms, because they have the same representation of the variables. b 9uv and 5vu?

Apply the idea Because u × v is always the same as v × u, we can combine terms that have the same variables multiplied by each other, even if the order in which they appear is different. They have the same representation of variables, so they are like terms.

Idea summary Two algebraic terms are called like terms if they have exactly the same variables and the variables have the same power. It doesn’t matter what order the variables are in.

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Combine like terms Like terms can be combined using addition or subtraction to simplify an expression. For example:

p + p = 2p This is called collecting like terms. To simplify the expression 2p + 3p, we can write out all the p’s: 2p + 3p = (p + p) + (p + p + p) =p+p+p+p+p = 5p It is much quicker to add the coefficients: 2p + 3p = 5p To simplify 2a + 3b + 4a, we can combine the like terms of 2a and + 4a. 2a + 3b + 4a = 6a + 3b To simplify 4x + 3y − 3x, we combine the x’s by solving 4x − 3x because there is a negative sign on the left of the 3x. It’s important that the negative or positive sign on the left of any term remains with it. 4x + 3y − 3x = x + 3y To simplify 9xy + 4b − 5xy + 2b, we have two pairs of like terms. We can combine 9xy and − 5xy and also 4b and + 2b. 9xy + 4b − 5xy + 2b = 4xy + 6b

Example 2 Which of these options correctly simpliﬁes the expression 4b + 3b? A 12b

7bb

Create a strategy

Apply the idea

We can add the terms all together by ﬁnding the sum of their coeﬃcients.

4b + 3b = (4 + 3)b

Find the sum of the

coeﬃcients = 7b

Evaluate the addition

The correct option is B: 7b.

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12b2

Example 3 Simplify 8x + 6y − 2y − 4x

Create a strategy To simplify an expression, collect all the like terms and subtract.

Apply the idea 8x + 6y − 2y − 4x = (8x − 4x) + (6y − 2y)

Collect like terms

= 4x + 4y

Evaluate the subtraction

Example 4 Simplify 35x ÷ 5

Create a strategy

Apply the idea

Divide the coefficient of the variable x by the second number.

Divide the coeﬃcient of x by 5

Evaluate the division

Idea summary Like terms can be combined using addition or subtraction to simplify an expression. It’s important that the negative or positive sign on the left of the term stays with the term.

8.04 Practice questions Understanding

Explorer 1–3

Adventurer 1, 3

Use the words “same” or “different” to complete:

Like terms have the ⬚ variables raised to the ⬚ power.

Trailblazer 1

Which statement is correct about the terms 2mn and 2nm? a

2mn and 2nm are the same because 2 × m × n is the same as 2 × n × m, as the order of the multiplication doesn’t matter.

2mn and 2nm are different because the order of the variables matters when multiplying.

Which statement is correct about the terms 5b and −3b? a

We cannot combine these terms because they have different coefficients.

We can combine these terms because they have the same variables. 8.04 Collect like terms mathspace.co

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Fluency Ex 1

7y and 6y

2 and −4

−13uv and 17vu

10z and 10b

8y and 5y

5ab and − 3cb

12mn and 18nm

11pr and 22rp

Write down any pairs of like terms in these expressions: a

10m + 2 + 8m + 3

9s − r − 6s + 1

k + 7j − 5 + 4k

10y + 4 + 5x + 3 − y

9 + 9a − 7b + 20 − 11a − 15b

2a − 3b + 9 − 10c

11mn − 15op − 8mn + 22

21 − 17ab − 3cd − 5ab + 7cd

9 + y + 10y − 9

10y + 1

9r + 8 − r + 1

10r + 2 − 2r + 7

9r + 8t

17rt

Are these expressions equal to 11y?

Ex 2

5y + 6y

9 + 8r

17r

Are these expressions equal to 9rt + 8? 5r + 4t + 8

8 + 4rt + 5rt

For these expressions: i

Name any pairs of like terms.

Simplify the expression.

5u + 7 + 8u + 8

4x − 6 + 5x

12 + 6x + 7y − 8

7u + 2v − 7u

Simplify these expressions where possible: a

2a + 5a

2b + 3b

7 + 6a

9 + 7m − 2p + 3q

12y − 3y + 1

4x + 7x − 5

3c + 4c + 7c

15x − 6x − 2x

19b − 12b − 6b

m 10ab + 3ab − 8ba n Ex 3

6y − 5y

Are these expressions equal to 8r + 9?

a 9

−3n + 6n + 3n

8x − 3x + 7x

4xy + 10xy − 9xy

15b − 7b + 10 + 1

9 + 7m − 2 + 3m

10x + 12 + 6x − 7

6p + 8q − 6p

Simplify: a

11m + 8n + 14m

7m + 8n + 3m + 2n

8x + 9y − 5x + 10y

7a + 11a + 9b − b

13m + 2n − 8m + 10n

2ab − 7c + ab + 9c

−3s + 6t + 9s − 4t

9xy + 20 + 12yx − 3

19a + 8b − 4c − 8a + b

8x + 6y − 2y − 4x + 2 + 3

10m + 7 + 9n + 5m + 10n

m 18x − 9y + 7x + 11y + 11 o Ex 4

314

4, 5bdfh, 7–9, 10j-p, 11h-p, 12e-l, 13

a 8

4, 5bdfh, 6–8, 9ab, 10c-m, 11a-i, 12a-d, 13

Describe these pairs of terms as like terms or unlike terms:

a 7

4, 5aceg, 6, 7, 9, 10a-i, 13

8x + 10 − 7y − 6z + 10z + 3

11y − 6z + y + 6z + 12 − 10

12j + 7 + 9k − 7k + 11j − 2

Simplify: a

x+x+x+x

2n + n + n

3m + 2m + m

4k + k + 3k

y×2

9×l

15 × h

21 × x

18z ÷ 3

36a ÷ 6

Mathspace WA Curriculum – Year 7 mathspace.co

Simplify: a

5x − 2y + 8x + 4 + 6z + 3y + 4z + 2

2xy + 5yz + 4xz + 8xy + 2xz − 3yz + 3

20 + 9x − 4y + 6y + 7x + 21

9mn − 8m − 7n + 7n + 10m + 2nm

11s + 6t + 3st − 2ts + t + 10 − 4s – 5

13ab − 8bc + 6ba − 9b + 11cb

Reasoning

14, 17

14, 16, 17

14–17

Why do you think we write y and not 1y when using algebra?

Are these terms like terms: 5abc and − 9acb? Explain your answer.

Substitute the values x = 5, x = 100 and x = 0.5 to show that 6x + 2x − x is equal to 7x.

Simplify: i

5x + 5x + 5x

4p + 4p + 4p + 4p + 4p

iii

−3c − 3c − 3c − 3c

0.5k + 0.5k + 0.5k

Review your results for part (a) and explain how the answer to 7 × 5m can be calculated.

Problem-solving 18, 20 18

19, 21, 22, 24

Complete these statements by filling in the blanks: a c e

19–21, 23

2x + ⬚ = 8x

7x − 5x + 10 − ⬚ = 2x + 1

7x + ⬚ + 10y − ⬚ = 11x + 6y

Consider the triangle shown.

b d f

3y + 3 + ⬚ + 5 = 8y + 8

9x + 4y + ⬚ = 9x + 6y

8ab + ⬚ + 9b + 6ab = ⬚ + 11b

Write an expression for the perimeter of the triangle in terms of x.

Simplify the expression.

10x − 3

3x + 9

x+6

Each side of a square is 3a metres in length. Calculate the distance around the outside of the square.

A library has a courtyard in the shape of an equilateral triangle with sides of length 5y metres. Calculate the distance around the outside of the courtyard.

Sam and Max each exercise for x minutes per day. Sam burns 8 calories per minute and Max burns 10 calories per minute. a

Write an expression for the number of calories Sam burns in one day.

Write an expression for the number of calories Max burns in one day.

Write a simpliﬁed expression for the total number of calories Sam and Max burn in one day.

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Ali, Ben and Mohammed each work for h hours per day. Ali earns $15 per hour, Ben earns $20 per hour and Mohammed earns $35 per hour. a

Write an expression for the amount Ali earns in one day.

Write an expression for the amount Ben earns in one day.

Write an expression for the amount Mohammed earns in one day.

Write a simpliﬁed expression for the total amount the three employees earn in one day.

If they work 8 hours in one day, how much do they earn altogether?

Sarah and Emily each read n pages per day. Sarah reads 15 pages per hour and Emily reads 20 pages per hour. a

Write an expression for the number of hours Sarah spends reading in one day.

Write an expression for the number of hours Emily spends reading in one day.

Write a simpliﬁed expression for the total number of hours Sarah and Emily spend reading in one day.

If they read a total of 120 pages, how many hours did they spend reading altogether?

Did you know?

Algebra helps us understand the branching patterns of trees! The way tree branches grow follows a mathematical principle known as branching fractals, where each branch splits into smaller branches at specific angles that can be described using algebraic formulas. By studying these patterns, scientists and engineers can apply algebra to model how trees and plants grow, which helps in fields like agriculture, forestry, and even urban planning. Algebra is the key to understanding how nature’s patterns evolve, from the trees in our forests to the plants in our gardens! 316

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8 Chapter review 1

Write a word statement for these expressions: a

z+4

Write an algebraic expression for these statements without using a multiplication (×) or division (÷) sign: a

Three times p plus 8

The difference of m and n, divided by 5

6 more than the quotient of 9 and r

(9 − l) ÷ 4

Simplify: a

2×w÷x−7

6+z×y−3

A courier company charges a flat fee of $15 plus $0.75 per kilogram for delivering parcels. Write an algebraic expression for the total cost of delivering a parcel that weighs p kilograms.

The length of a rectangle is 6 less than twice its width. Write an algebraic expression for the length of the rectangle in terms of its width, x.

The sum of the ages of a father and his son is 60 years. If the father’s age is r years, write an algebraic expression for the son’s age.

Find the value of: a

4y − 5 when y = 6

when x = 2 and z = 5

3b2 + 8 when b = 3

p2 + qr when p = 4, q = 3 and r = 6

u2 + v3 + w when u = 5, v = 2 and w = 9

Complete these tables: a

1 6

r+5 b

2 7

1 4

4r r 5r + 3 10

Find the value of: a

2 8 1 8

3 8

3 12

3 18

2 13

When a whole number is substituted for y in the expression 7y + 2, the result is a number greater than 65. What is the smallest value that the whole number could be?

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If s is an integer greater than 0 and less than 8, find the minimum possible value of 12 − 3s.

Lisa breeds cats for sale. Each of her c cats gives birth to k kittens. If she has 4 cats, write an expression for the total number of kittens in terms of k.

Simplify these expressions where possible: a

5xz + 12xz − 7xz

15cd + 6cd − 11dc

15x − 4y + x + 4y + 8 − 6

3 + 9p + 3q − 5p + 8q − 9

Simplify: a

3 + 8b

26 × 2x

81a ÷ 6

Simplify: a

15r + 8u + 4ru − 3ur + u + 12 − 6r − 7

11cd − 7de + 8dc − 10d + 13ed

A park has a square-shaped lawn with sides of length 4x metres. Write an expression for the distance around the entire lawn in terms of x.

Jenny and Tony each exercise for y minutes per day. Jenny burns 6 calories per minute, while Tony burns 9 calories per minute. Write a simplified expression for the total number of calories Jenny and Tony burn in one day in terms of y.

Sara, Alex and Jane each work for k hours per day. Sara earns $25 per hour, Alex earns $30 per hour and Jane earns $40 per hour. Write a simplified expression for the total amount Sara, Alex and Jane will earn per day in terms of k.

318

John and Karen each read p pages per day. John reads 10 pages per hour and Karen reads 15 pages per hour. a

Write an expression for the total number of hours John and Karen read in one day in terms of p.

Using the expression from part (a), calculate the total number of hours they spend reading if they each read 180 pages.

Mathspace WA Curriculum – Year 7 mathspace.co

“Everything should be made as simple as possible, but not simpler.” Albert Einstein

Big ideas Creating, solving and manipulating equations helps us to model the relationships in the world around us.

Curriculum Outcomes • Solve simple linear equations involving up to two operations and verify the solution by substitution • Explore, describe and represent concrete and real-world, linear and non-linear growing patterns using a table of values and a graph. Determine unknown values in the pattern

9 Equations Chapter outline 9.01 9.02 9.03 9.04

Introduction to equations Write rules for relationships One-step equations Two-step equations Chapter 9 review

322 329 338 343 353

Think of a linear equation as a balanced scale. Just as both sides of the scale have the same weight, both sides of the equation have the same value.

9.01 Introduction to equations After this lesson, you will be able to… • define and identify the key components of an equation, including left-hand side, right-hand side and the equals sign. • determine if an equation is true or false based on the equality of its expressions. • solve equations with a single unknown variable to find the solution that makes the equation true. • convert word problems into algebraic equations and vice versa.

Equations An expression is a statement, such as 6 + 4 or 28 ÷ 14. Expressions can sometimes be evaluated and in these cases we add an equals sign. Doing this turns our expression into an equation. 6 + 4 = 10 and 28 ÷ 14 = 2 are both equations. An equation is a way of saying that two mathematical expressions are equal to each other. We can think of equations as balanced scales. The equation is 6 + 4 = 10

This equation consists of a left-hand side, 6 + 4, an equals sign = and a right-hand side, 10. This equation is true because 6 + 4 does equal 10. An equation is true when the left-hand side is equal to the right-hand side.

Example 1 Is 7x − 8(x + 9) an equation or an expression?

Create a strategy An equation is a sentence. It has something on the left-hand side, an = symbol and something on the right-hand side. An expression is a phrase that represents a number. It does not have an = symbol.

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Apply the idea Since 7x − 8(x + 9) doesn’t contain an = symbol, it is an expression.

Example 2 Is this equation true or false?

Create a strategy Evaluate the expressions on either side of the equals sign.

Apply the idea Evaluate the left-hand side (LHS):

Evaluate the right-hand side (RHS): RHS = 106 − 94 = 12 We have found that both sides are equal to 12. is true.

The equation

Reflect and check Remember that a fraction can be treated the same as division, so

is the same as 96 ÷ 8.

Idea summary An equation joins two expressions together using an equals sign. An equation is true if the expression on the left-hand side is equal in value to the expression on the right-hand side.

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Make equations true When we ﬁnd the value of an unknown in an equation that makes the equation true, it is called solving an equation. Consider this equation: 20 + ⬚ = 24

We know that the unknown number to make the equation true is 4. We call 4 the solution of the equation. A solution to an equation is a value that makes it true. In algebra we usually use variables to represent unknowns, so the equation could be written as 20 + x = 24 The solution is x=4

Example 3 Solve the equation for s. s − 7 = 14

Create a strategy This is asking what value of s will make the equation true. s needs to be 7 more than 14.

Apply the idea

Reﬂect and check

s = 14 + 7

Add 7 to 14

Does the value of s = 21 make the equation true?

s = 21

Evaluate

s − 7 = 21 − 7 = 14 This is correct, so s = 21 does make the equation true.

Idea summary The number that makes an equation true is called the solution. The unknown in an equation can be represented by a variable like x, J or s.

Convert between words and equations Equations are often used to solve problems involving words. We can translate words into mathematical equations and vice-versa.

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Example 4 Describe the equation

using words.

Create a strategy Translate each mathematical symbol into words.

Apply the idea 45

Forty-five

divided by

is equal to

three

So a description for the equation

is:

Forty-five divided by x is equal to three.

Reﬂect and check It should be noted that there is more than one way to describe an equation. In this case, the description could also be: The quotient of forty-five and x is equal to three.

Example 5 Which of these equations matches the description ‘the product of eight and s is taken away from sixty-six. The result is equal to ten’? A 8(66 − s) = 10

66 − 8s = 10

8s − 66s = 10

8s − 66 = 10

Create a strategy The word ‘product’ means the answer to a multiplication. The phrase ‘taken away’ indicates subtraction while ‘the result is equal to’ means the same as.

Apply the idea The phrase ‘the product of eight and s’ means ‘eight multiplied by s’. The phrase ‘the product of eight and s is taken away from sixty-six’ is the same as ‘sixty-six minus eight times s’. Convert the words directly into mathematical symbols: Sixty six 66

minus −

eight 8

times ×

s s

is equal to =

ten 10

So we have the equation: 66 − 8s = 10 Option B is the correct answer.

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Idea summary We can translate words into equations and equations into words.

9.01 Practice questions Understanding 1

Explorer 1–4

Adventurer 1–4

Trailblazer 2, 4

Choose the correct word for each blank: An ⬚ (equals sign/operator) joins two ⬚ (expressions/equations) together using an equals sign.

An ⬚ (expression/equation) is true if the expression on the left-hand side is ⬚ (added/equal) to the expression on the right-hand side. 2

What is the difference between an expression and an equation?

What does it mean to ‘solve’ an equation?

Paul solved the equation x + 5 = 11 and found x = 6. How can Paul check that his solution is correct?

Fluency Ex 1

3x + 5 = 14

2x + 3

4z = 12

2 − 5y

Ex 3

326

5, 6d-h, 7, 8, 9cdf, 10cf-h, 11be-h, 12–14, 16

2(7 + 4z)

⬚ + 18 = 35

2y − 7 = 3y + 1

Fill in the blank to make these equations true: a

5, 6a-e, 7, 8, 9abe, 10a-e, 11b-f, 13ab, 14, 16

Which of these are equations and which are expressions?

e Ex 2

5, 6a-d, 8, 9acd, 10a-c, 11a-df, 15

4+⬚=8

8 × ⬚ = 35 − 3

b f

3 × ⬚ = 30

⬚÷3=9−6

c g

⬚ × 8 = 42 − 10

⬚ × 11 = 77

36 ÷ ⬚ = 15 − 6

Determine whether these equations are true or false. If false, make a change to one of the values to make it true. a

8 + 16 = 6 × 4

5 × 7 − 19 = 13

57 – 4 = 15 + 6 × 7

Solve: a

x + 18 = 30

8x = 96

28 − x = 21

12 + m = 25

f − 9 = 14

11u = 66

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 4

Ex 5

Write a corresponding word sentence that describes the mathematical relationship for each equation: a

8x = 32

5y = 60

6x + 4 = 60 − 20

8x + 7 = 72 − 25

Write an equation that represents each word sentence: a

The quotient of g and seven is equal to eleven.

Sixteen less than m is equal to twenty six.

Twelve less than q is equal to twenty ﬁve.

The product of seven and y is equal to x less than twenty ﬁve.

The quotient of p and ten is equal to seven.

The product of ﬁve and q is equal to p less than twenty nine.

The product of seven and p is taken away from sixty ﬁve. The result is equal to twelve.

The product of twelve and r is taken away from seventy two. The result is thirty six.

Identify if p = 8 is a solution to each equation: a

23 + p = 31

29 + p = 35

35 − p = 27

p − 14 = 6

p − 11 = −3

6p = 48

11p = 96

13p = 104

Identify whether the given value of x is a solution to each equation: a

17 + x = 40, x = 20

c e

x − 30 = 21, x = 48

2x = 10, x = 5

8 − x = 10, x = 2

7x = 33, x = 5

g 13

Identify if the given solution for each equation is correct: a

u + 3.3 = 4.8, u = 2.2

v + 20.7 = 51.41, v = 20.71

Identify if y = 2.2 is a solution to each equation: a

2.3 + y = 4.5

3.2 − y = 1.2

5y = 1.2

4y = 8.08

4.5 − y = 3.2

y + 3.5 = 4.5

Consider the expression m − 17. a

Complete the table: m

m − 17 b

What value of m will make the equation m − 17 = 12 true?

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Consider the expression 4 × k. a

Complete the table: k

4×k b

What value of k will make the equation 4 × k = 96 true?

Reasoning 17

17, 18

Consider the equation 56 − t = 39. a

Which side of the equation is bigger when substituting t = 10?

Should we try a smaller or larger number than 10 to solve the equation?

Which side of the equation is bigger when substituting t = 20?

What can be said about the solution of the equation?

Consider the equation 2u = 70. a

Which side of the equation is bigger when substituting u = 30?

Should we try a smaller or larger number than 30 to solve the equation?

Which side of the equation is bigger when substituting u = 40?

What can be said about the solution of the equation?

Problem-solving 19 19

17, 18

19, 20

19–22

A company charges $95 per hour for their services. If a customer uses their services for x hours and pays $855, write an equation to ﬁnd the hourly rate. Which equation would match the scenario? A

855x = 95

x + 95 = 855

95x = 855

There are 20 cupcakes on the table. Soleil can bake 12 cupcakes per hour. Write an equation to ﬁnd the number of cupcakes (C) per hour (h).

The price of a kilogram of apples is p dollars. If John bought 10 kilograms of apples for $30, write an equation that matches the scenario.

Bob is currently b years old. In 16 years’ time his age will be 42. Write an equation to represent Bob’s current age.

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9.02 Write rules for relationships After this lesson, you will be able to… • identify and analyse different representations of patterns, such as number patterns, rules, equations and tables of values. • convert a pattern’s rule into an equation by using variables and simplifying the equation. • use substitution to find any number in the pattern using the equation. • construct different representations of the relationship between a pattern and its equation.

Patterns in tables Patterns can be represented in lots of different ways: a number pattern, a rule, an equation or a table of values. Consider this example: Daniel is aiming to run a half-marathon. In the ﬁrst week he plans to run three kilometres, then each week he plans to run three kilometres further than the week before. We can display this pattern in a table, where: w = week number (top row) d = distance ran in kilometres (bottom row) Notice that the top numbers in the table are increasing by 1 each time. 1

The bottom numbers in the table are increasing by 3 each time. We can see that the next two numbers after 15 are 18 and 21.

If we compare the top numbers and the bottom numbers we see that the distance is always three times the week number. To write this as an equation we rewrite it using numbers and variables. The rule is: distance = 3 × week number d = 3w 9.02 Write rules for relationships mathspace.co

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Consider another table of values: 1

This time the y-values are increasing by 5 as the x-values increase by 1. This means the rule will contain ×5.

To ﬁnd the rule between the x and y-values use the 5 and think “5 × x + ? = y” 5×1+1=6 5 × 2 + 1 = 11 5 × 3 + 1 = 16 5×x+1=y The rule is: y = x times 5 then add 1 y = 5x + 1 Consider another table of values: +2

In this table the x-values are increasing by 2 and the y-values are increasing by 4. In other words, the y-values change by 2 when the x-values change by 1.

The rule will have × 2 in it. The rule is: y is two times x y = 2x

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Example 1 Peter is making a sequence of shapes out of matchsticks.

a Peter makes a table comparing the number of triangles to the number of matchsticks needed to construct it. Number of triangles Number of matchsticks

1 3

2 5

3 7

5 11

10 21

20 41

Which of these statements describes the relationship between the number of matchsticks and the number of triangles? A

The number of matchsticks is twice the number of triangles.

The number of matchsticks is twice the number of triangles plus one.

The number of triangles is one more than twice the number of matchsticks.

There is no relationship between the number of matchsticks and the number of triangles.

Create a strategy Compare the two rows in the table. Identify the common difference between the number of triangles and the number of matchsticks.

Apply the idea We can see that the number of matchsticks is always one more than twice the number of triangles. So the correct statement that describes the relationship between the number of matchsticks and the number of triangles is option B.

b Using t to represent the number of triangles and m to represent the number of matchsticks, which of these equations describes the relationship between the number of matchsticks and the number of triangles? A

m = 2t + 1

m = 2t − 1

m = 2t + 2

m=t+2

Create a strategy We can convert our solution from part (a) into a mathematical equation to ﬁnd our solution.

Apply the idea We know from part (a) that the number of matchsticks is always one more than twice the number of triangles. We can replace the words with their mathematical symbols to find the equation. So the correct equation that describes the relationship between the number of matchsticks and the number of triangles is option A.

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Example 2 Vanessa opens a bank account and deposits $300. At the end of each week she adds $10 to her account. a Complete the table to show the balance of Vanessa’s account over the ﬁrst four weeks. Week (W ) Account total (A)

Create a strategy Find the account total using the starting balance and deposits.

Apply the idea If Vanessa starts with $300 and adds $10 to her account at the end of every week then the account total should start at $300 for week 0 then increase by $10 each week after. Week (W ) Account total (A)

0 $300

1 $310

2 $320

3 $330

4 $340

b Which of these equations represents the relationship between Vanessa’s account total (A) and the number of weeks (W ) for which she has been adding to her account? A

A = 300

A = 300 + 10W

A = 300W + 10

A = 10W

Create a strategy Use the table in part (a) and ﬁnd how much each variable is changing by.

Apply the idea Week (W ) Account total (A)

0 $300

1 $310

2 $320

3 $330

4 $340

We can see that the values of W go up by 1 and the values of A go up by 10, so we need to multiply W by 10. But 0 × 10 = 10 and we need 310, so we need to add the initial 300. The correct answer is option B.

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Idea summary Tables of values can be used to display patterns. To ﬁnd the equation for a pattern in a table of values, in the form y = mx + c, where m and c are numbers to be found: • m is the change in y when x increases by 1 • c is the value of y when x = 0 So for: 1

• m = 5 as this is the change in y when x increases by 1 • c = 1, we can work out the value of y when x = 0 by subtracting 5 from the value of y when x = 1. For this pattern the equation is: y = 5x + 1

9.02 Practice questions Understanding 1

Explorer

Adventurer 1cd, 2

Trailblazer 1cd, 2

Use words to describe the pattern in the y-values in these tables. For example: “increasing by 2”. a

x 1 2 3 y 5 6 7

4 8

5 9

x y

6 12

7 10

8 8

9 6

10 4

x 1 2 3 y 7 14 21

4 28

5 35

x y

1 100

2 95

3 90

4 85

5 80

The pattern in the y-values of a table can help us find the rule between the x and y-values. If the x-values are increasing by 1 and the y-values are increasing by 7 then the coefficient of x in the rule will be 7. Identify the coefficient of x for the rule in a table of values where: a

x is increasing by 1 and y is increasing by 3.

x is increasing by 1 and y is decreasing by 2.

x is increasing by 2 and y is increasing by 6.

x is increasing by 2 and y is decreasing by 8. 9.02 Write rules for relationships mathspace.co

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Fluency 3

3–6, 8ace, 9ab

Figure 2

Figure 3

Complete the table: Figure number (F ) Number of matchsticks (M )

By how many matchsticks is the number increasing each time?

Write the rule for the number of matchsticks, M, in terms of the ﬁgure number, F.

Hiro is making a sequence of shapes out of matchsticks: Figure 2

Hiro makes a table comparing the ﬁgure number to the number of matchsticks needed to construct it as shown:

Figure 3

Figure 4

Figure number (F ) Matchsticks (M )

1 5

2 6

Complete the table for the above pattern.

By how many matchsticks is the number increasing each time?

Write the algebraic rule for the number of matchsticks, M, in terms of the ﬁgure number, F, for this pattern.

Dave is constructing a continuing pattern of squares using matchsticks: Figure 1

Figure 2

Figure 3

Dave wants to continue the pattern. Determine the number of matchsticks he needs to add for each extra square.

Dave made a table comparing the ﬁgure number to the number of matchsticks required. Describe any pattern that you notice.

Complete the statement: The number of matchsticks is equal to ⬚ times the ﬁgure number plus ⬚.

d 334

Figure 1

4, 6, 7, 8bdf, 9bdf

Yvonne has constructed the ﬁrst three triangles of a pattern using matchsticks: Figure 1

Ex 1

4–7, 8ace, 9ace

Figure no. (F ) Matchsticks (M )

1 4

2 7

3 10

4 13

Write the algebraic rule for the number of matchsticks, M, in terms of the ﬁgure number, F.

Mathspace WA Curriculum – Year 7 mathspace.co

Vanessa is making a sequence out of tiles. She creates a table to record the ﬁgure number of a shape and the number of tiles needed to make it:

Figure 1

Figure 2

Figure number (F )

No. of tiles (N )

Figure 3

Write an equation for the number of tiles, N, in terms of the ﬁgure number, F.

Calculate the number of tiles required for F = 12.

James is making snowflakes out of hexagonal tiles:

James creates a table comparing the width of a snowﬂake to the number of tiles needed to make it: Width (W ) No. of tiles (T )

1 1

3 7

5 13

7 19

Write an equation for the number of tiles, T, in terms of the snowﬂake’s width, W.

Calculate the number of tiles, T, when, W = 100.

For these tables: i

Find the change in y for each increase in x by 1.

Write an equation for y in terms of x.

x 1 2 3 4 y 9 10 11 12

5 13

x y

6 1

7 2

8 3

9 4

10 5

x 1 2 3 4 y 7 14 21 28

5 35

x y

1 7

2 12

3 17

4 22

5 27

x 1 2 3 4 y 14 11 8 5

5 2

x y

3 1

5 5

7 9

9 13

11 17

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For each table of values, write an equation for y in terms of x: a

x 6 7 8 9 y 3 4 5 6

10 7

x y

10 50

11 55

12 60

13 65

14 70

x 6 7 8 9 y 13 14 15 16

10 17

x y

5 12

6 14

7 16

8 18

9 20

x 5 7 9 11 y 2 8 14 20

13 26

x y

72 8

81 9

90 10

99 11

108 12

Reasoning 10

10, 11

Explain how to ﬁnd the missing value in this table using two different methods: a b

4 6

5 10

6 14

8 22

Is the rule y = 5x + 2 the same as the rule y = 2 + 5x? Explain your thinking.

Consider the pattern of triangles shown with the rule M = 3T, which translates to “the number of matchsticks is equal to three times the number of triangles”.

Explain the signiﬁcance of the number 3 in the rule.

Write a rule for a similar pattern made with squares.

Write a rule for a similar pattern made with hexagons.

Problem-solving 13, 14, 16 Ex 2

10–12

15–17

Gwen sells bananas in bunches of 4. a

Complete the table: No. of bunches (x) No. of bananas (y)

b c

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13, 15, 16

Complete the statement: The number of bananas is equal to ⬚ times the number of bunches.

Write an equation for the number of bananas, y, in terms of the number of bunches, x.

Mathspace WA Curriculum – Year 7 mathspace.co

Quentin buys some decks of playing cards that contain 52 cards each. a

Complete the table: No. of decks (d) No. of cards (c)

b c 15

Complete the statement: The number of cards is equal to ⬚ times the number of decks. Write an equation for the number of cards, c, in terms of the number of decks, d.

Complete the statement: The number of marbles will be equal to ⬚ times the number of bags plus his original ⬚. Write the algebraic rule for the number of marbles Liam will own, m, in terms of the number of bags he buys, b.

Kendra opens a bank account and deposits $200. At the end of each week, she adds $15 to her account. a

Complete this table which shows the balance of Kendra’s account over the ﬁrst four weeks: Week (W ) Account total (A)

Liam already owns 5 marbles. He then buys some bags of marbles containing 4 marbles each. a

0 $200

1 $215

Write the algebraic rule for Kendra’s account total, A, in terms of the number of weeks W, for which she has been adding to her account.

Vanessa makes the ﬁrst four entries of a sequence out of coloured tiles as shown:

Figure 1

Figure 2

Figure 3

She then creates a table to show the relationship between the ﬁgure number and the number of tiles used for that ﬁgure:

Figure 4

Figure number No. of tiles used

1 1

Complete the table of values.

What type of numbers are in the No. of tiles used row?

Write the algebraic rule for the number of tiles, N, in terms of the ﬁgure number, F, for this pattern.

How many tiles would be needed for ﬁgure number 7 for this pattern?

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9.03 One-step equations After this lesson, you will be able to… • use inverse operations to solve equations.

Inverse operations An equation can be thought of as a set of balanced scales. This image shows the equation x + 12 = 20 x

+ 12

+ 20

To ﬁnd the value of x, we need to subtract 12 from the left-hand side of the scales. To keep the scales balanced we also need to subtract 12 from the right-hand side of the scales. We write x + 12 − 12 = 20 − 12 Simplifying both sides gives x=8 To solve the equation, we subtracted 12 from both sides, which is the opposite of adding 12. This is called using inverse operations. • Addition and subtraction reverse each other • Multiplication and division reverse each other When solving an equation, the goal of using inverse operations is to have the variable on one side of the equals sign by itself, with the value of that variable on the other side.

Example 1 Consider the equation 9p = 36. a Which of these options will result in p being isolated on one side of the equals sign? A

Divide both sides by 9

Subtract 9 from both sides

Multiply both sides by 0

Multiply both sides by 9

Create a strategy What operation is being done to p? It is being multiplied by 9. So we need to know what the inverse operation of ×9 is.

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Apply the idea The inverse operation for ×9, is ÷9. So the correct answer is Option A.

b Apply the operation from part (a) to the equation to ﬁnd the solution.

Create a strategy Divide both sides of the equation by 9.

Apply the idea Write the equation

Divide both sides by 9

Evaluate

Example 2 Solve the equation v − 4 = 44.

Create a strategy The reverse of subtracting 4 is adding 4. So we need to add 4 to both sides of the equation.

Apply the idea v − 4 = 44

Write the equation

v − 4 + 4 = 44 + 4 Add 4 to both sides v = 48

Evaluate

Reflect and check Substitute the value of v back into the equation to see if it is correct. v − 4 = 48 − 4 = 44

Example 3 Solve the equation 24 + p = 59.

Create a strategy The reverse of adding 24 is subtracting 24. So we need to subtract 24 from both sides of the equation.

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Apply the idea 24 + p = 59

Write the equation

p + 24 = 59

Swap the position of p and 24

p + 24 − 24 = 59 – 24 Subtract 24 from both sides p = 35

Evaluate

Example 4 Solve:

Create a strategy A fraction is a way of representing a division process. The reverse of division is multiplication, so we need to multiply both sides of the equation.

Apply the idea Write the equation

Multiply both sides by 8

Evaluate

Idea summary To solve an equation we need to apply inverse operations to both sides of the equation. Inverse operations reverse each other: • Addition and subtraction are inverse operations • Multiplication and division are inverse operations

9.03 Practice questions Understanding 1

Adventurer 1, 2

Complete these statements: a b

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Explorer 1, 2

⬚ and subtraction are inverse operations. ⬚ and division are inverse operations.

Mathspace WA Curriculum – Year 7 mathspace.co

Trailblazer 2

Match the equation with the correct inverse operation. i

Add 8

Subtract 8

iii

8x = 72

x + 8 = 20

3a-c,g, 4ab, 5ab, 6ab, 7ab, 8a-i, 10a-h

Fluency Ex 1

3, 4cd, 5cd, 6cd, 7cd, 8h-p, 9, 10el-n

s − 2 = 35

t + 3 = −45

−4.5 + x = 6.8

4.1x = 94.3

x − 1.39 = 8.67

4n = 27.2

36.5 = 7.3n

Use inverse operations to solve: x − 15 = 35

y−7=2

b − 8 = 27

k − 14 = −7

b + 10 = 72

k + 8 = −10

5x = 45

9a = 63

Use inverse operations to solve: 14 + x = 64

12 + y = 21

Use inverse operations to solve: 3x = 90

2d = 24

Use inverse operations to solve: b

a 8

x−8=5

a Ex 4

7p = 63

a 6

Multiply by 8

a Ex 3

3a-h, 4bc, 5bc, 6bc, 7bc, 8c-i, 10i-k

Identify the inverse operation that should be applied to both sides in order to solve each equation:

i Ex 2

Divide by 8

Solve: a

q + 8 = 10

s + 8 = 29

r + 22 = 69

u − 3 = −4

v − 4 = 44

u − 22 = 4

4k = 20

8k = 56

y + 5 = −6

24 + p = 59

−48 + x = 30

−27 + m = −32

−10.22 + x = −3.62

m 22 − x = 22

12n = 60

Solve: a

x + 5.6 = 7.4

x + 16 = 16

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For each statement: i

Write an equation, where p is the unknown number.

Solve the equation.

When 10 is added to a number, the result is 12.

When −9 is added to a number, the result is 21.

When 20 is subtracted from a number, the result is 5.

When 13 is subtracted from a number, the result is −15.

The sum of an unknown number and 39 is 83.

75 is equal to the sum of an unknown number and 42.

17 less than an unknown number is 25.

The quotient of an unknown number and 8 is 12.

When a number is multiplied by 10, the result is 60.

When a number is divided by 8, the result is 9.

The product of 3 and an unknown number is 18.

When a number is multiplied by 12, the result is 84.

m When a number is divided by 11, the result is 11. n

7 multiplied by an unknown number is 84.

Reasoning

11, 12

12, 13

If x + 5 = 21 and you want to ﬁnd x, would you add or subtract 5 from both sides of the equation? Why?

Consider the equation x − 8 = −5. Is it possible to solve this equation by adding 8 to both sides? Why or why not?

Samantha is planning to buy a new laptop that costs $900. She has saved $600 so far and plans to earn the remaining amount by doing freelance work.

342

Samantha charges $30 per hour for her freelance work. Write an algebraic equation to represent the number of hours (h) she needs to work to earn the remaining amount needed to buy the laptop (r).

How many hours does Samantha need to work to earn the remaining amount needed to buy the laptop?

Samantha’s friend, Lisa, also wants to buy the same laptop but has already saved $780. How many hours of freelance work does Lisa need to do to earn the remaining amount needed to buy the laptop? Assume Lisa gets paid at the same rate as Samantha.

What are some reasons why Samantha might charge more or less per hour for her freelance work compared to Lisa?

Mathspace WA Curriculum – Year 7 mathspace.co

Problem-solving 14 14

14, 15

Write an equation and then solve: a

A bag of lollies contains 80 lollies. The lollies are shared evenly between 10 children. How many lollies (n) will each child receive?

Justin divides a deck of cards into 7 piles. There are 12 cards in each pile. How many cards (c) are in the deck?

Tricia gives Tom 25 marbles from her collection. Tricia now has only 44 marbles in her collection. How many marbles (m) did Tricia have initially?

Lisa bought a hat and a dress for a total of $80. Find the cost of the hat (h) if the cost of the dress is $56.

A company has a total of 500 employees, of which 60% are over 35 years old and the rest are 35 years and under. a

If the company plans to increase its under 35 workforce by 25%, how many new under 35 year old employees should they hire to achieve this goal?

What percentage of the total workforce will be 35 and under after the new hires?

9.04 Two-step equations After this lesson, you will be able to… • use flowcharts to analyse the structure of a two-step equation, identifying the operations used to construct the equation. • determine the inverse operations required to solve a two-step equation based on the flowchart analysis. • apply the inverse operations in the reverse order to both sides of the equation in order to solve for the variable. • solve a variety of two-step equations, demonstrating the ability to apply the learned techniques effectively and accurately.

Flow charts A ﬂow chart can help us understand how an equation has been built and what inverse operations, in what order are required to solve the equation. Consider the equation 6y + 9 = 33 Looking at the left-hand side, we can determine what the order of operations are, that is, what is happening to the y in order.

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343

y is ﬁrst multiplied by 6 and then we add 9: 6 y

9 6y 9

To undo these operations we must backtrack. We must apply the inverse operations in the reverse order to return to the variable y: 6 y

9 6y 9

6y 6

This tells us that to solve the equation we must ﬁrst subtract 9 from both sides and then divide both sides by 6. If we were to solve the equation using the inverse operations we just identified, it would look like this: Subtract 9 from both sides

Divide both sides by 6

Evaluate

Example 1 By following the operations given, complete the entries. 4

Create a strategy Multiply 8 by 4 for the ﬁrst step, then add 1 for the second.

Apply the idea 4 8

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1 32

Example 2 Identify the inverse operations and then backtrack to complete the entries. 5

16 41

Create a strategy Identify the inverse operations for each step and write them underneath the flow chart. After that, use the inverse operations to fill in the blanks.

Apply the idea 5

41 5

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Example 3 For 7x − 24 = 18: a Create a flow chart.

Create a strategy Create a flow chart of the left-hand side to identify the order of operations.

Apply the idea 7 x

24 7x 24

b Solve by identifying the inverse operations.

Create a strategy Use the flow chart from part (a) to identify the inverse operations.

Apply the idea Determine the inverse operations. 7 x

24 7x 24

7x 7

Now solve using the inverse operations.

Idea summary A ﬂowchart can be used to determine what operations in what order will be required to solve an equation.

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Solve 2-step equations A ﬂow chart helps us understand how an equation is built. Now we will apply the inverse operations in the reverse order to both sides of an equation in order to solve the equation. Consider the equation: 4x + 7 = 31 It has been built starting with x and then the operations ‘multiply by 4’ and ‘add 7’. To solve for x, we must ﬁrst do ‘subtract 7’ and then ‘divide by 4’ to both sides of the equation. Subtract 7 Divide by 4

4x + 7 = 31 4x = 24 x=6

Subtract 7 Divide by 4

Here is another example: 7(y + 6) = 49 It has been built starting with y then ‘add 6’, then ‘multiply by 7’, The inverse operations in reverse order to solve the equation will be: 1. First divide both sides by 7. 2. Then subtract 6 from both sides. Divide by 7 Subtract by 6

7 ( y + 6) = 49 y+6=7 y=1

Divide by 7 Subtract by 6

Example 4 Solve the equation

Create a strategy Apply the inverse operations in the reverse order.

Apply the idea We can see from the equation that the expression on the left-hand side was built by applying the operations ‘add 7’ and ‘divide by 2’ to the variable u. So the order of reverse operations will be: 1. Multiply by 2 2. Subtract 7 Write the equation

Multiply both sides by 2

Evaluate

Subtract 7 from both sides

Evaluate

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Example 5 Solve the equation 3(s − 2) = 21.

Create a strategy Apply the inverse operations in the reverse order.

Apply the idea We can see from the equation that the expression on the left-hand side was built by applying the operations ‘subtract 2’ and ‘multiply by 3’ to the variable s. So the order of reverse operations will be: 1. Divide by 3 2. Add 2 Write the equation

Divide both sides by 3

Evaluate

Add 2 on both sides of equation

Evaluate

Example 6 Solve the equation

Create a strategy Apply the inverse operations in the reverse order.

Apply the idea We can see from the equation that the expression on the right-hand side was built by applying the operations ‘divide by 5’ and ‘subtract 3’ to the variable t. The order of reverse operations will be: 1. Add 3 2. Multiply by 5 Write the equation

Add 3 to both sides

Evaluate

Multiply both sides by 5

Evaluate

Swap sides

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Idea summary To solve a multiple-step equation, ﬁrst identify the operations used to build the equation. Then apply the inverse operations in the reverse order to both sides of the equation.

9.04 Practice questions Explorer

Understanding 1

Put socks on then put shoes on.

Get into bed then turn lamp off.

Turn on TV then start watching movie.

Take lunch box out of bag then start eating lunch.

×5 then + 4 is reversed by doing −4 then ÷ 5. Write the operations required to reverse these operations in the correct order: ×3 then + 11

−8 then × 6

÷5 then − 2

+5 then ÷ 9

×2

÷5

Identify the inverse operation: +3

−7

4–7, 9, 10, 11ac, 12a-ch, 14

Fluency 4

1–3

Ex 1

Trailblazer

1–3

Write the steps of these actions in the reverse order:

a 3

Adventurer

1–3

4–6, 8–10, 11ad, 12b-e

4–6, 9, 11bc, 12d-g, 13

Follow the operations to complete the entries in the flow chart: 2

3 2 4

3 2

(6 3) 2

c 8 Ex 2

Identify the inverse operations and then backtrack to fill in the boxes: 4

a 7

b 9

2 26

2 14

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Ex 3

For each equation: i

Create a flow chart of the left side of the equation to identify the order of operations.

Solve by identifying the inverse operations.

5y + 10 = 20

e 7

Ex 6

8y + 9 = 1

The answer is 12. Write this as an equation.

Write the two inverse operations required to solve this equation in the correct order.

I have a number, k. I add 8 and then divide the result by 5. Write this as an expression.

The answer is 6. Write this as an equation.

Write the two inverse operations required to solve this equation in the correct order.

I have a number, x. I divide it by 2 and then add 5. Write this as an expression.

The answer is 19. Write this as an equation.

Write the two inverse operations required to solve this equation in the correct order.

Solve: a

2w + 9 = 37

4x + 9 = 25

7m + 18 = 39

6y + 4 = 34

13x − 2 = 24

5t − 15 = 15

29m − 8 = 21

2p − 17 = 3

Solve:

Solve: a

5(t + 9) = 60

2(p + 14) = 58

9(x + 2) = 27

6(x + 5) = 54

5(n − 15) = 35

3(s − 16) = 21

9(p − 3) = 0

4(s − 8) = 16

−13 = 7z − 48

48 = 6(p + 8)

−17 + 8s = 7

Solve:

3(x − 12) = 90

32 = 8(2 + s)

−15 + 5x = 10

Soledad has x marbles. After receiving 18 marbles from her brother Kiko, her classmate Rhian gave her additional marbles so that her number of marbles was doubled. Soledad now has 72 marbles. a

Complete the ﬂow chart to represent the situation. 18

b 350

6(s − 9) = 24

I have a number, n. I multiply it by 2 and then add 6. Write this as an expression.

a Ex 5

5(q + 3) = −10

i Ex 4

How many marbles did Soledad have initially?

Mathspace WA Curriculum – Year 7 mathspace.co

Reasoning

16–18

Explain why subtracting 3 is the incorrect first step to solving 21 = 7(m + 3).

Write an equation involving p, division and subtraction, that has a solution of p = 8.

Consider this equation where A and B are positive integers, less than 12. Ax + B = 12 Write the reverse operations required to solve the equation for x.

Consider this equation where C, D and E are positive integers. C (x − D) = E a

Write the reverse operations required to solve the equation for x.

Write the equation with x as the subject.

Problem-solving 20

19, 20, 24

19–26

A taxi charges a ﬂat rate of $10 plus $2 for each kilometre driven. If the total cost of the ride was $98, how many kilometres did the taxi travel?

A gardening service charges a base fee of $50 and an additional $15 for each hour of work. If the total cost for a gardening job was $230, how many hours did the gardener work on the job?

Jenna has a balance of $1500 in her account. Each month, she makes a deposit of $40. Additionally, she has made a one-time withdrawal of $20. How many months did it take for Jenna to accumulate her current balance, assuming she started with a balance of $0?

A social media account gains 400 followers every week but loses 1000 followers due to inactivity. If the account currently has 17 000 followers, how many months did it take to reach this number?

A charity organisation has a current fund of $1120. They receive monthly donations of $150 and spend $80 on one of their charitable activities. How many months did it take for the charity organisation to reach their current fund balance?

The area of a rectangle is given by the formula A = length × width. a

If the length of a rectangle is 9 m and its width is 3 m, ﬁnd its area.

The area of a particular rectangle is 45 m2. If its length is 9 m, determine the width of the rectangle.

Width

Length

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The perimeter of a triangle can be calculated with the formula P = a + b + c, where a, b and c are the three side lengths of the triangle. If a triangle has a perimeter of P = 37 cm and side lengths a = 14 cm and b = 13 cm, ﬁnd the length of the side c.

The formula to convert temperature from Celsius to Fahrenheit is a

If C = 35, ﬁnd the value of F.

If F = 212, ﬁnd the value of C.

Did you know?

Equations play a vital role in healthcare! For example, doctors and pharmacists use formulas that factor in body weight and age to calculate the correct dosage of medicine for each patient. These calculations help ensure treatments are both safe and effective, reducing the risk of side effects or complications. Equations are also used to track how medication moves through the body, manage chronic conditions like diabetes, and even fine-tune hospital equipment. Maths allows health professionals to make informed, life-saving decisions — turning numbers into better care for everyone.

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9 Chapter review 1

Determine whether these equations have x = 12 as a solution: a

25 − x = 13

17 + x = 39

Determine whether these equations have y = 3.5 as a solution: a

x − 24 = 12

6y = 21

A company charges $115 per hour for their services. If a customer uses their services for x hours and pays $1035, which equation would match this scenario? A

1035x = 115

x + 115 = 1035

115x = 1035

Billy is b years old. In 12 years time his age will be 39. Write an equation using b that could be used to solve for Billy’s current age.

For each table of values, write an equation for y in terms of x: a

x 1 3 5 7 y 5 11 17 23

x y

2 13

3 21

4 29

5 37

6 45

Identify the inverse operation that should be applied to both sides in order to solve each equation: a

3x = 342

x + 8.5 = 25.6

x − 5 = 29

x + 7 = 48

4x = 164

x + 14 = 31

8x = 112

Use inverse operations to solve: a

x + 5 = 35

11x = 176

x − 9 = 47

Solve: a

−15 + x = 28

−19 + x = −43

9x = 117

x + 13 = 31

−12.2 + x = −2.8

Solve: a

9 29

For each statement: i

Write an equation, where n is the unknown number.

Solve the equation.

15 multiplied by an unknown number is 225.

19 less than an unknown number is 76.

The quotient of an unknown number and 12 is 24. Chapter 9 review mathspace.co

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Write an equation and then solve: a

A box of chocolates contains 90 chocolates. The chocolates are distributed evenly amongst 6 friends. How many chocolates (c) will each friend receive?

Samantha splits a bundle of pencils into 5 equal groups. There are 7 pencils in each group. How many pencils (p) are in the bundle?

Alex gives Lisa 20 books from his library. Alex now has only 30 books in his library. How many books (b) did Alex have initially?

Solve: a

4x − 2 = 14

5m + 15 = 30

7n − 21 = 0

5 (x − 4) = 0

3 (y − 6) = 9

7 (z − 5) = 56

Consider the pattern. a

Describe how the pattern changes over each step.

Determine the number of circles in the next step of the pattern.

2c + 10 = 16

Solve: a

4b + 3 = 19

Solve: a

5a + 7 = 22

Step 1

Step 2

Step 3

Step 4

Write the algebraic rule for the number of circles, C, in terms of the step number, S, for this pattern.

James has y books. After receiving 12 books from his sister, his friend Nathan gave him additional books so that his number of books was tripled. James now has 54 books. How many books did James have initially?

A shuttle bus charges a flat rate of $15 plus $3 for each kilometre driven. If the total cost of the ride was $120, how many kilometres did the shuttle bus travel?

The area of a kite is given by the formula

, where p and q are the lengths of the

diagonals. A particular kite has a short diagonal length p = 7 m and area A = 42 m2. Find the value of q.

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Janella makes the first four entries of a sequence out of groups of circles as shown. She then creates a table to show the relationship between the figure number and the number of circles Figure 1 used for that figure: Figure number No. of circles

1 3

2 7

Figure 2

Figure 3

Figure 4

Complete the table for the above pattern.

By how much does the number of circles required to make each figure increase?

Write the algebraic rule for the number of circles, C, in terms of the figure number, F, for this pattern.

The perimeter of a rectangle is given by the formula P = 2 × (l + w), where l is the length and w is the width. If a certain rectangle has length l = 27.5 cm and perimeter P = 81.4 cm, find the width w.

The formula to convert temperature from Fahrenheit to Celsius is a

The area of a rhombus is given by the formula x and y are the lengths of the diagonals. a b

If F = 68, find the value of C.

If C = 15, find the value of F. , where

Find the area of a rhombus which has short and long diagonal lengths of 4 cm and 6 cm respectively. A particular rhombus has a short diagonal length x = 6 m and area A = 33 m2. Find the value of y.

y x

The perimeter of a rectangle is given by the formula P = 2 × (l + w), where l is the length and w is the width. a

Find the perimeter of the rectangle which has a length of l = 6 cm and width of w = 5 cm.

If a certain rectangle has length l = 7 cm and perimeter P = 24 cm, ﬁnd the width w.

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Big ideas The Cartesian plane provides a visual representation of the relationship between two quantities by plotting them as ordered pairs on a set of perpendicular axes.

Curriculum Outcomes • Plot coordinates on the Cartesian plane and explore, visualise, predict and determine image coordinates after translation or reflection across the axes or rotation about the origin • Explore, describe and represent concrete and real-world, linear and non-linear growing patterns using a table of values and a graph. Determine unknown values in the pattern

10 Cartesian plane Chapter outline 10.01 The Cartesian plane 10.02 Directed numbers on the Cartesian plane 10.03 Graphs of linear relationships Chapter 10 review

358 366 376 386

GPS navigation systems use the Cartesian plane to help you find your way!

10.01

The Cartesian plane

After this lesson, you will be able to… • plot points on the Cartesian plane and write their coordinates. • find the distance between two points on the Cartesian plane.

The Cartesian plane We can think of the Cartesian plane as being built from two separate number lines. The x-axis is the horizontal number line and it is perpendicular (at right angles) to the vertical y-axis. Every point plotted in the Cartesian plane has an x-coordinate and a y-coordinate. In the image, the cat has coordinates (6, 2). The x-coordinate or horizontal distance, is 6 and the y-coordinate or vertical distance, is 2.

(6, 2)

2 1 0

x 1

Coordinates are always written with brackets in the form (x, y) where the ﬁrst number, x, is the x-coordinate and the second number, y, is the y-coordinate. The point at which the x-axis and the y-axis intersect has coordinates (0, 0) and is called the origin.

Interactive exploration

Discover this concept in action online

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Mathspace WA Curriculum – Year 7 mathspace.co

mathspace.co

Example 1 Here is a Cartesian plane.

9 8 7 6 5 4 3 2 1

a What object has coordinates (6, 2)?

x 1

2 3 4 5 6 7 8 9 10

Create a strategy Use the numbers on the axes to find the coordinates.

Apply the idea Start at (0, 0). Move 6 spaces to the right, then 2 spaces up. The object with coordinates (6, 2) is a bike. b Write the coordinates of the rabbit.

Create a strategy Count the number of horizontal and vertical units required to move away from the origin and determine the location of the rabbit.

Apply the idea The rabbit is located 9 spaces to the right, then 8 spaces up. So, the coordinates are (9, 8).

Example 2 Plot the point (6, 3) onto the Cartesian plane.

Create a strategy Use the numbers on the axes to move on the coordinate plane.

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Apply the idea Start at (0, 0). Plot the point 6 spaces to the right, then 3 spaces up. This will be the point on the plane described by (6, 3). 9

8 7 6 5 4 3 2 1 0

x 1

Example 3 Write the coordinates of the point that is 5 units to the right of (9, 6).

Create a strategy Plot the given coordinates, then move horizontally by the required number of units.

Apply the idea 10

Plot (9, 6) on the Cartesian plane and move 5 units to the right.

The new coordinates are (14, 6).

8 7 (9, 6)

6 5

(14, 6) 5 units

4 3 2 1 0

x 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reﬂect and check Another way to ﬁnd the coordinates of the new point is by realising that moving right will increase the x-coordinate so we need to add 5 to the x-coordinate. New coordinates = (9 + 5, 6) = (14, 6)

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Add 5 to 9 Evaluate

Example 4 Point A has the coordinates (3, 6) and point B has the coordinates (8, 6). What is the distance between A and B?

Create a strategy Plot the points on the coordinate plane and then count the horizontal units from point A to point B.

Apply the idea Plot the points A and point B. The distance between them is 5 units. 9

8 7 A

5 units

5 4 3 2 1 0

x 1

Reﬂect and check Another way to ﬁnd the distance between the two points is by realising that point A and point B have the same y-coordinate, therefore the distance between them is the difference in their x-coordinate. Distance = (8 − 3) =5

Subtract 3 from 8 Evaluate

Idea summary The Cartesian plane is used to describe points in a two-dimensional space. These points are labelled by coordinates (x, y). The horizontal line is called the x-axis and the vertical line is called the y-axis. The point at which the x-axis and the y-axis intersect is called the origin, which has coordinates of (0, 0).

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10.01 Practice questions Understanding 1

Explorer

Adventurer

1, 2

Trailblazer

1–3

Which of these coordinates have an x-coordinate of 2? (2, 5), (6, 2), (2, 0), (1, 2)

Which of these coordinates have a y-coordinate of 3? (3, 10), (3, 0), (0, 3), (6, 3)

Identify whether each statement is true or false: a

The coordinate plane is used to describe the location of points in a two-dimensional space.

The number line laying down horizontally on the plane is the y-axis.

The vertical distance along the y-axis is the y-coordinate.

The coordinates of the point at which the x-axis and the y-axis intersect are (0, 0).

Fluency Ex 1a

4–6, 7a-h, 8, 10, 13–15, 16ac

4–6, 7c-m, 8–10, 11a-d, 13–15, 16ac, 17

Using the Cartesian plane: Identify the object that has these coordinates: a

(1, 4)

(4, 3)

(2, 6)

(9, 10)

y 10 9 8 7 6 5 4 3 2 1 0

Ex 1b

362

Using the Cartesian plane: Write the coordinates of these objects: a

Kite

Car

Teddy bear

Bird

Mathspace WA Curriculum – Year 7 mathspace.co

4–5, 7i-p, 8–12, 15–19

1 2

x 3 4 5 6 7 8 9 10

1 2

3 4 5 6 7 8 9 10 x

y 10 9 8 7 6 5 4 3 2 1 0

Write down the coordinates of these points: a

y F

8 7 6 5

B D E

4 3 2

1 1 Ex 2

A 2 3 4 5 6 7 8 9

Plot these points on a Cartesian plane: a

A (3, 0)

B (0, 8)

C (5, 6)

D (6, 2)

E (7, 7)

F (0, 0)

G (3, 5)

H (7, 2)

I (0, 6)

J (9, 3)

K (7, 6)

L (5, 9)

m M (8, 8)

N (8, 2)

O (9, 0)

P (0, 2)

Consider the three points: A (3, 5), B (2, 8) and C (4, 2) a

Plot these points on a Cartesian plane.

Which point is closest to the x-axis?

Which point is closest to the y-axis?

Consider the Cartesian plane: a

Write the coordinates for points A, B, C, D and E.

For all of the plotted points, is the x or y-coordinate larger?

What does the x-coordinate and y-coordinate of each point add up to?

y 10 A 9 B 8 C 7 D 6 E 5 4 3 2 1

1 2 3 4 5 6 7 8 9 10

Write down the coordinates of the vertices of each shape: a

y 9

9 8

C x

2 3 4 5 6 7 8 9

x 1

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Ex 3

Find the coordinates of these points: a

The point six units to the right of the origin.

The point four units above the origin.

The point four units to the right of (9, 6).

The point seven units to the left of (7, 8).

The point one unit to the left and eight units below the point (3, 10).

The point three units to the right and two units above the point (2, 5).

Which of these points is furthest from the origin? A

Ex 4

(0, 3)

(0, 10)

(6, 0)

(9, 0)

Consider the point with coordinates (4, 8). How many units above the origin is this point located?

Consider the point with coordinates (2, 7). How many units to the right of the origin is this point located?

Consider the point A (4, 6). Find the coordinates of point A if it moves:

4 units to the right.

4 units to the left.

4 units upward.

4 units downward.

Find the distance between these two points: a

A (3, 6) and B (9, 6)

C (10, 2) and D (3, 2)

P (7, 5) and Q (7, 9)

R (9, 12) and S (9, 4)

Consider the points: A (2, 9), B (10, 9) and C (10, 5) a

Plot the points on a Cartesian plane.

Find the length of AB.

Find the length of BC.

A triangle has vertices with coordinates A (5, 9), B (1, 2) and C (7, 2). a

Plot the triangle ABC on a Cartesian plane.

Find the perpendicular height of the triangle if BC is the base.

Find the length of the base BC.

A quadrilateral has vertices with coordinates P (3, 8), Q (6, 8), R (8, 5) and S (1, 5). a

Plot the quadrilateral PQRS on a Cartesian plane.

Find the length of PQ.

Find the length RS.

Identify what type of quadrilateral PQRS is.

Reasoning 20

364

21–22

20a-c, 22

20cd, 22–23

Describe in words how to get to point A from point B on the Cartesian plane for each pair of points: a

A (0, 0), B (7, 0)

A (0, 0), B (0, 10)

A (3, 6), B (8, 2)

A (9, 8), B (3, 4)

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Consider the Cartesian plane: a When following gridlines, describe in words how to get:

from the book to the car.

from the beach ball to the burger.

Opie wants to place a picture of a cat 7 units to the right and 4 units up from the star. Which of the objects currently on the Cartesian plane will it be closest to?

Consider the Cartesian plane. If each side of a square on the grid is 1 metre and we can only travel along the gridlines, determine the shortest distance: a from the teddy bear to the star. b

from the beach ball to the apple.

from the burger to the beach ball, collecting the star and the apple on the way.

1 2

3 4 5 6 7 8 9 10 x

1 2

3 4 5 6 7 8 9 10 x

y 10 9 8 7 6 5 4 3 2 1 0

Point A (a, b) and point B (c, d) are two points on the Cartesian plane. If a < c and b > d describe in words how to go from point A to point B.

Problem-solving 24 24

y 10 9 8 7 6 5 4 3 2 1 0

24–25

A fire hose is turned on. In the graph, the y-axis shows the height (metres) of the stream of water and the x-axis shows the horizontal distance from the hose. The hose is positioned at the origin.

24–26 y

12 m

Write down the coordinates of point A.

10 m

Write down the coordinates of point B. Write down the coordinates of point C.

c d

The water reaches its maximum height at point C. Find the horizontal distance of point C from the hose. What is the maximum height reached by the water?

14 m

16 m

4m 2m

0m 1m 2m 3m 4m 5m 6m 7m 8m

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The number of people that logged onto a new website over a 12-hour period was recorded every hour by the graph:

1000 800

How many people were logged on when the site initially went live?

600

How many people were logged on 6 hours after the site went live?

400

After how many hours was the greatest number of people on the site recorded?

What was the greatest number of people recorded on the site?

People

200 0

Hours 2

The profile of a skateboard ramp is shown on the graph. y A competitor in a competition starts at the point A, 8m A rides down the slope through point B and is at ground level at point C. a

When the competitor is at point B: i ii

ow far have they travelled horizontally from H the starting point?

ow far have they travelled vertically from H the starting point?

How far will they have travelled horizontally from point A the next time they are at the same height as point B?

B x C 1m 2m 3m 4m 5m 6m 7m

When performing a trick, competitors aim to be 1 metre above the end of the ramp. How far above the ground is this?

10.02 Directed numbers on the Cartesian plane After this lesson, you will be able to… • identify the quadrant that a given point belongs to. • understand the concept of negative coordinates on the Cartesian plane. • identify and plot coordinates in each quadrant.

Quadrants on the Cartesian plane In the previous lesson, we worked with a Cartesian plane that had only positive coordinates. If we extend the axes in both directions, we can locate points with negative coordinates. 366

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Interactive exploration

Discover this concept in action online

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The Cartesian plane is divided into four distinct regions, called quadrants. y Quadrant 2

Quadrant 1

0 Quadrant 3

x Quadrant 4

The 1st quadrant is on the top right. The x-coordinate and y-coordinate of a point in the 1st quadrant are both positive. The quadrants are numbered in an anticlockwise direction: • 2nd quadrant: x-coordinates are negative, y-coordinates are positive • 3rd quadrant: both coordinates are negative • 4th quadrant: x-coordinates are positive, y-coordinates are negative Points that lie on an axis, like (−5, 0) or (0, 4), are not in any quadrant. Points on the x-axis have a y-coordinate of 0. Points on the y-axis have an x-coordinate of 0.

Example 1 What are the coordinates of the points A, B and C?

4 A

3 2 1

−5 −4 −3 −2 −1 −1

x 1

2 3 4 5 C

−2 −3 B −4 −5

Create a strategy The coordinates are in the form (x, y), where x is the value on the x-axis (horizontal) and the y is the value on the y-axis (vertical).

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Apply the idea For point A: • The x-coordinate is -5 and y-coordinate is 3. So the coordinates of point A is (-5, 3). For point B: • The x-coordinate is -2 and y-coordinate is -4. So the coordinates of point B is (-2, -4). For point C : • The x-coordinate is 3 and y-coordinate is -1. So the coordinates of point C is (3, -1).

Example 2 In which quadrant does the point (3, −2) lie?

Create a strategy

Apply the idea

Recall the characteristics of each quadrant:

Since the coordinates have positive x and negative y, the point (3, −2) lies in 4th quadrant.

• 1st quadrant: positive x and positive y. • 2nd quadrant: negative x and positive y. • 3rd quadrant: negative x and negative y. • 4th quadrant: positive x and negative y.

Reﬂect and check We can plot the point to check which quadrant it is in. Quadrant 2

Quadrant 1

3 2 1 −4 −3 −2 −1

−1

−2

x 1

4 (3, −2)

−3 Quadrant 3

The point is in Quadrant 4.

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−4

Quadrant 4

Example 3 What is the distance between A (6, 8) and B (−3, 8)?

Create a strategy Since the y-coordinates are the same, find the difference of the x-coordinates.

Apply the idea Distance = 6 - (-3)

Subtract -3 from 6

=6+3

Combine the adjacent signs

= 9 units

Evaluate

Reﬂect and check We can plot both points on the plane and then see if we can determine the distance. If the points are in a horizontal or vertical line the distance is the difference between the x or y-values. We can see that the points are in a horizontal line. The total distance is 3 units on the left of the y-axis and 6 units on the right of the y-axis, which is a total of 9 units.

y (−3, 8)

(6, 8)

8 6 4 2

−8 −6 −4 −2 −2

x 2

−4 −6 −8

Idea summary The coordinate plane is divided into 4 quadrants. Points that lie on an axis, like (−5, 0) or (0, 4), are not in any quadrant. Points on the x-axis have a y-coordinate of 0. Points on the y-axis have an x-coordinate of 0.

y Quadrant 2

Quadrant 1

0 Quadrant 3

Quadrant 4

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10.02 Practice questions Understanding 1

Adventurer 1–3, 5–8

the origin

any point on the y-axis

any point on the x-axis

Identify the y-coordinate of: i

the origin

If point P has an x-coordinate of 0, which axis must it lie on?

If (a, b) is a point that lies on the x-axis, ﬁnd the value of b.

Identify the quadrant(s) that have:

Points with a negative y-coordinate.

Points with a negative x-coordinate.

Points where the x-coordinate and y-coordinate have the same sign.

Points with a negative x-coordinate and a positive y-coordinate.

Points with a negative x-coordinate and a negative y-coordinate.

Find the coordinates of these points given the description: a

The point 9 units below the origin.

The point 4 units to the left of (−3, 6).

The point 7 units to the right of (−1, −2).

The point 2 units to the right and 2 units below the point (2, 5).

The point 6 units to the left and 5 units above the point (4, −4).

The point 3.5 units to the left of the origin.

Which point is furthest from the origin? A

(0, −3)

(5, 0)

(0, 4.5)

How many units above the origin is the point (−6, 1) located?

Starting at the origin, move 4 units left and then 4 units up.

370

Trailblazer 1–4, 5de, 6–8

Identify the x-coordinate of: i

Explorer 1–2, 5–6, 8

Plot the resulting point on a Cartesian plane.

Identify the coordinates of this point.

Mathspace WA Curriculum – Year 7 mathspace.co

(−4, 0)

9a-i, 10–12, 13a-j, 14–15, 16ac, 17ab, 18ac

Fluency

Ex 1

9d-m, 10–12, 13e-n, 14–15, 16bd, 17bd, 18bd, 19

9g-p, 10–12, 13k-t, 14, 16bd, 17cd, 18bd

Consider the Cartesian plane: H

F E

9 8 7 6 5 4 3 2 1

B A D

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1 K −2 −3 −4 −5 −6 −7 J I −8 L −9

Identify the coordinates of these points: a

m M

Consider the given points on the Cartesian plane: Identify which point has these coordinates:

10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4 S −6 −8 −10

(9, −3)

(−4, −7)

(2, 4)

(−8, 8)

P x 2 4 6 8 10 Q

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Consider the Cartesian plane: a

Identify the coordinates of these points:

3 2 B −4 −3 −2 −1

x 1

−1

−2 −3

−4

Consider the points A and B plotted on the Cartesian plane:

Write down the coordinates of point A.

Write down the coordinates of point B.

Which axis do points A and B lie on?

1 B −4 −3 −2 −1

−1

x 4

−2 −3 −4

Plot these points on a Cartesian plane: a

S (1, 0)

T (0, 2)

N (0, 8)

O (1, 3)

G (2, 6)

A (6, 1)

B (−7, 3)

C (7, −4)

D (−7, −3)

E (−4, −4)

F (−6, 0)

H (2, −8)

m I (−6, 3)

J (−5, −5)

K (−5, 9)

L (5, −9)

M (3, −3)

P (−4, 5)

Q (−2, −4)

R (4, −3)

(2, −1)

(−5, −6)

Identify which quadrant the point lies in.

q Ex 2

In which quadrant do these points lie? a

372

(−3, 4)

For each point: i

(7, 8)

Plot the point on a Cartesian plane.

(2, 5)

(−2, 1)

(−3, −5)

(8, −5)

(−8, −1)

(4, −7)

(0, 2)

(5, 0)

For each set of points: i

Plot the points on a Cartesian plane.

Identify the shape formed by joining the points in order.

(4, 1), (7, 1), (4, 4), (7, 4)

(1, 5), (6, 5), (1, 8), (6, 8)

(−5, −3), (0, −1), (−4, 6), (1, 8)

(−4, −3), (0, −1), (−2, 1), (2, 3)

Mathspace WA Curriculum – Year 7 mathspace.co

For each shape, write down the coordinates of the vertices: a

y D 10 8 6 4 2

−10 −8 −6 −4 −2 −2 C −4 −6 −8 −10

−10 −8 −6 −4 −2 −2 −4 −6 C −8 −10

T x

2 4 6 8 10

10 8 6 4 2

−10 −8 −6 −4 −2 −2 U −4 −6 −8 −10

2 4 6 8 10

A (−5, 8) and B (−2, 8).

A (7, 3) and B (−1, 3).

A (6, −5) and B (6, −1).

A (−6, 2) and B (−6, −7).

Consider the points: A (−4, 8), B (−7, 8) and C (−7, 1). a

Plot the points on a Cartesian plane.

Find the length of BC. 20–21

Find the length of AB.

20–21

Describe how you would move on the Cartesian plane from the origin, to plot these points: a

Plot these pairs of points on a Cartesian plane and then find the distance between them:

Reasoning 20

−10 −8 −6 −4 −2 −2 −4 −6 −8 V −10

2 4 6 8 10 B

10 8 6 4 2

Ex 3

b A

(2, −4)

(9, 14)

Describe how you would move on the Cartesian plane from point A to point B. a

A (5, −2), B (−3, 1)

A (−5, −7), B (−1, 3)

A (4, −2), B (10, −7)

A (3, −5), B (−3, 6)

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Problem-solving 22ab, 23–24 22

22bd, 23–25

22cd, 23–27

Consider this Cartesian plane. 4

If the squares of the grid have side lengths of 1 m and we can only move along gridlines, find the shortest distance for these routes. a

From point A to point C

From point C to point D

From point C to point D, going via point B

From point A to point D, going via points B and C

y A

3 2 B −4 −3 −2 −1

x 1

−1

The points given represent three vertices of a rectangle. Find the coordinates of the fourth vertex if it is known to be in the second quadrant.

The points given represent three vertices of a rhombus. Find the coordinates of the fourth vertex if the missing point lies in the third quadrant.

−4

5 4 3 2 1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

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−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−2 −3

2 3 4 5

The points given represent two of the three vertices of a triangle. a

Identify three possible coordinates of point C if the three points form an isosceles triangle.

Identify four possible coordinates of point C if the three points form a right-angled triangle.

The points form an isosceles trapezium.

The points form a parallelogram with point D in the second quadrant.

The points form a parallelogram with point D in the third quadrant.

The points form a parallelogram with point D in the first quadrant (this will be beyond the grid shown).

−5 −4 −3 −2 −1 −1 −2 −3 B −4 −5

Consider the three points on the Cartesian plane. Identify the coordinates for point D for each of the situations below. a

5 4 3 2 1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 C −2 −3 −4 −5

2 3 4 5

A 1

2 3 4 5 B

A triangle has points A (1, 2), B (−2, −3) and C (6, −3). a

Plot the triangle ABC on a Cartesian plane.

Find the perpendicular height of the triangle if BC is the base.

Find the length of base BC.

Find the area of triangle ABC.

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10.03 Graphs of linear relationships After this lesson, you will be able to… • plot points from a table of values and recognise that each column in a table of values corresponds to an ordered pair of coordinates. • identify linear relationships. • graph relationships from tables of values.

Plot points from a table of values Each column in a table of values corresponds to a pair of coordinates, also called an ordered pair. Consider this table of values with the rule y = x + 3. To complete the table of values we evaluate y, for each value of x in the table using the rule y = x + 3. For example, when x = 0, y = 0 + 3 = 3. x y

−2 1

−1 2

0 3

1 4

2 5

The table contains 5 ordered pairs: (−2, 1), (−1, 2), (0, 3), (1, 4), (2, 5) y

2 1

1 −3

−2

−1

x 1

−1

The 5 points can be plotted on a Cartesian plane.

−3

−2

−1

The points form a straight line which means that y = x + 3 is a linear relationship.

A linear relationship is formed when the coordinates form a straight line or are close to a straight line.

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Example 1 Consider the equation y = 3x. a Complete the table of values. x y

−5

−3

−1

Create a strategy Substitute each x-value into the equation.

Apply the idea For x = -5, y = 3 × (-5) = -15

For x = -3, Substitute x = −5

= -9

Evaluate

For x = -1, y = 3 × (-1) = -3

y = 3 × (-3)

Substitute x = −3 Evaluate

For x = 1, Substitute x = −1

y = 3 × (1) =3

Evaluate x y

−5 −15

−3 −9

−1 −3

Substitute x = 1 Evaluate

1 3

b Plot the points from the table of values.

Create a strategy

Apply the idea

Each column in the table of values represents a coordinate pair. Plot the coordinates on the Cartesian plane.

3 2 1 −6 −5 −4 −3 −2 −1

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14 −15

y x 1

The coordinates from the table are: (−5, −15), (−3, −9), (−1, −3), (1, 3).

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c Is the relationship linear?

Create a strategy

Apply the idea

Use a ruler to see if a straight line can be drawn through all points.

3 2 1 −6 −5 −4 −3 −2 −1

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14 −15

y x 1

A straight line can be drawn through all points so the relationship is linear.

Example 2 Create a table of values identifying the number of matchsticks for each figure and create a graph of the relationship. Is this relationship linear? Figure 1

Figure 2

Figure 3

Create a strategy Complete the table of values. Use the figure number as the x-value and the number of matchsticks as the y-value. Plot the values on a Cartesian plane.

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Apply the idea x y

1 4

2 7

3 10

Because the points are all in a straight line, the relationship is linear. This means that the pattern grows at a constant amount with each added figure.

y 10 8 6 4 2

x 1

Idea summary Each column in a table of values represents a pair of coordinates. Each coordinate pair can be plotted on a Cartesian plane. If we can draw a straight line that passes through all the plotted points, we say the table of values and the equation that describes it represents a linear relationship.

Did you know?

Video games use the Cartesian plane to create virtual worlds! Game developers use the x- and y-axes to determine the positions of characters and objects on the screen, while the z-axis adds depth, creating the 3D experience. Game physics, such as collision detection and jumping mechanics, also rely on these axes to function accurately in the virtual world.

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10.03 Practice questions Understanding 1

Trailblazer 1–3

x −3 −2 −1 y 7 5 3

0 1

x y

−2 −2

−1 0

0 2

2 4

Do the points plotted on each Cartesian plane form a linear or non-linear relationship? a

5 4 3 2 1

8 7 6 5 4 3 2 1 −4 −3 −2 −1

−1 −2 −3

y 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Adventurer 1–3

List the four coordinates displayed in each table of values. a

Explorer 1–2

2 3 4 5

−11−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −1

−2 −3 −4 y

x 1

Identify four points from the graph that form a linear relationship.

y 4 3 2 1 B S O −4 −3 −2 −1 1 2 R −1 A

−2 −3 −4

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M x 3

Fluency 4

Ex 1

4a-c, 5ac, 6, 8

Plot the points from the table on a Cartesian plane.

Determine whether the points form a linear relationship.

x 1 2 3 4 y 1 2 3 4

x y

0 0

1 1

4 2

9 3

x 1 2 3 4 y −4 −3 −2 −1

x y

−4 0

−3 1

−2 2

−1 3

x −2 −1 0 1 y 4 1 0 1

x y

−2 −8

−1 −4

0 0

1 4

−1

Complete the table of values provided.

Plot the points from the table on a Cartesian plane.

y = 4x −1

y=x−5 x y

y = x + 3 x y

2 8

For each equation:

4d-f, 5bd, 7–10

For each table of values:

x y

4b-d, 5bd, 6–7, 10–11

y = 2x + 7

Consider the given table of values: a

Plot the points in the table on a Cartesian plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 4, find the corresponding value of y.

Consider the given table of values: a

Plot the points in the table on a Cartesian plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 8, find the corresponding value of y.

Consider the given table of values: a

Plot the points in the table on a Cartesian plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 3, find the corresponding value of y.

x y

−2

x y

0 0

1 5

2 10

3 15

x y

−8 −2

−4 −1

0 0

4 1

x y

−1 −3

0 1

1 5

2 9

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A farmer sells apples by weight. The farmer charges $5 per kilogram of apples. a

Fill in the table of values to show the total cost for different weights of apples. Weight (Kg) Total Cost ($)

Plot the points from the table on a Cartesian plane.

Is the relationship linear or non-linear?

Let the height of a pot plant be y cm. As the pot plant grows, the height increases according to the equation y = 2t + 9, where t is the elapsed time in weeks. Complete the table of values: Time (t weeks) Height of pot plant ( y cm)

Ex 2

Plot the points from the table on a Cartesian plane.

Is the relationship linear or non-linear?

Diane is making a sequence of shapes out of matchsticks. Figure 1

Figure 2

Figure 4

Complete the table of values by identifying by the number of matchsticks for each figure. Figure number (x) Number of matchsticks ( y)

Figure 3

Plot the points in the table on a Cartesian plane.

Reasoning

12–13

12–14

12–15

What is the minimum number of points you need to be able to graph a linear relationship?

Consider the three points A (3, 3), B (5, 5) and C (7, 7). Without plotting the points, name three other points that lie on the same line formed by these points.

Consider the three points X (−3, −7), Y (−3, −9) and Z (1, −2). Without plotting the points, explain whether the points form a linear relationship.

Consider the given table of values: Does the ordered pair (−3, −5) satisfy the linear relationship between x and y?

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x y

−2 −2

−1 1

0 4

1 7

Problem-solving 16, 18–19 16

Complete the table of values: Week No. of apples

17–18, 20–23

After each week, the number of apples on an apple tree increases according to the diagram: a

16, 18–22

Week 1

Week 2

Week 3

Week 4

Plot the number of apples against each week on a Cartesian plane.

Is the relationship between the number of weeks and the number of apples linear?

If the relationship between the number of weeks and the number of apples is linear, write the equation for the relationship where x is the number of weeks and y the number of apples.

Consider the given table of values for a linear relationship. a

Find the value of y when x = 0

Find the value of y when x = 2

Find the value of y when x = −5

Find the value of x when y = 13

x y

−3 −11

−2 −8

−1 −5

This pattern was created using matchsticks. Each figure is labelled with a number: a

Complete the table of values: Figure number No. of matchsticks

Plot the table of values on a Cartesian plane.

Is the relationship between the figure number and the number of matchsticks linear?

If the relationship between the figure number and the number of matchsticks used is linear, write the equation for the relationship, where x is the figure number and y the number of matchsticks.

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The points on the Cartesian plane show the relationship between water temperature, x, and surface air temperature, y:

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−3

−2

−1

Find the surface air temperature when the water temperature is 4 °C.

Find the surface air temperature when the water temperature is 14 °C.

Write the rule for the relationship between the surface air temperature and the water temperature.

A staircase has 3 steps where each step has a height of 15 cm and a length of 10 cm. a

Complete the table of values: No. of steps Height of staircase (cm)

1 15

10 cm 15 cm

Plot the height of the staircase against the number of steps on a Cartesian plane.

Is the relationship between the height of the staircase and the number of steps linear?

If the relationship between the height of the staircase and the number of steps is linear, write the equation for the relationship, where x is the number of steps and y is the height of the staircase.

Is it possible to extend the staircase to exactly 105 cm high, with steps that have the same height and length as in the image?

Complete the table of values: No. of steps Length of staircase (cm)

384

2 3 4 5

Complete the table of values: Water temperature in °C (x) Surface air temperature in °C ( y)

1 10

Plot the length of the staircase against the number of steps on a Cartesian plane.

Is the relationship between the length of the staircase and the number of steps in a staircase linear?

Is it possible to have a staircase 70 cm long, with steps that have the same height and length as in the image?

Mathspace WA Curriculum – Year 7 mathspace.co

A bakery records the number of loaves sold each day over a week and the total revenue generated from the sales. For a sale of 2 loaves, the bakery earns $4. For a sale of 3 loaves, the bakery earns $6 and for 5 loaves, the bakery earns $10. a

Complete the table of values with the number of loaves sold and the total revenue for each day. Number of loaves, l Revenue, r

l=2

r = 2l

l = 2r + 2

Plot the points from the table on a Cartesian plane.

Is the relationship linear?

Determine the bakery’s revenue if they sell 4 loaves of bread.

l=r+2

Let the capacity of water in a tank be C L and the time taken to fill up the tank be t minutes. Suppose the tank is initially empty and the number of litres in the tank is always three times the number of minutes that have passed. a

Complete the table of values: Time (t mins) Capacity of water (C L)

Plot the points from the table on a Cartesian plane.

Which of these equations represents the relationship between the time taken, t, and the capacity of the tank, C? A

d 23

Which equation represents the relationship between the number of loaves and the revenue? A

C=3

C = 3t

C = 3t + 2

C=t+3

Is the relationship between time taken and capacity of water linear or non-linear?

Max is 21 years old and Sebastian is 14 years old. Let M represent Max’s age and S represent Sebastian’s age. a

Write an equation for M in terms of S.

Identify whether these ordered pairs, in the form (S, M), satisfy the relationship between Max and Sebastian’s ages:

(14, 21)

(36, 29)

iii

(24, 31)

(16, 23)

vii (38, 46)

(20, 27)

(25, 25)

Plot the points from part (b) that satisfy the relationship between their ages on a Cartesian plane.

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10 Chapter review 1

If point Q has a y-coordinate of 0, which axis must it lie on?

If (a, b) is a point that lies on the y-axis, find the value of a.

In which quadrant do these points lie? a

(3, −2)

(−8, −3)

(−4, 7)

(0, 6)

(−6, 0)

Which point is furthest from the origin? A

(9, 5)

(0, −5)

(7, 0)

Find the coordinates of these points: a

The point eight units horizontally to the right of the origin.

The point five units vertically above the origin.

The point five units horizontally to the right of (12, 7).

The point nine units horizontally to the left of (15, 10).

The point two units horizontally to the left and six units vertically below the point (4, 11).

The point four units horizontally to the right and three units vertically above the point (3, 6).

Consider the point B (6, 8). Find the coordinates of point B if it moves: a

5 units horizontally to the right

5 units horizontally to the left

5 units vertically upward

5 units vertically downward

A quadrilateral has vertices with coordinates A (2, 7), B (5, 7), C (7, 4) and D (4, 4). a

Plot the quadrilateral ABCD on a Cartesian plane.

Find the length of AD.

Find the length of BC.

Identify what type of quadrilateral ABCD is.

Consider this Cartesian plane. 4

Identify the coordinates of these points: a

2 B

−4 −3 −2 −1

−1

x 1

−2 −3 C

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−4

Consider the given points on the Cartesian plane. 10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4 S −6 −8 −10

P x 2 4 6 8 10 Q

Identify which point has these coordinates:

(6, −2)

(−3, −5)

(3, 5)

(−7, 7)

For each set of points: i

Plot the points on a Cartesian plane.

Identify the shape formed by joining the points in order.

(3, 2), (6, 2), (3, 5), (6, 5)

(−3, −2), (1, 0), (−1, 2), (3, 4)

Find the distance between these pairs of points: a

A (−6, 9) and B (−3, 9)

A (8, 4) and B (0, 4)

A (7, −6) and B (7, −2)

A (−7, 3) and B (−7, −6)

Consider the given table of values: a

Plot the points from the table on a Cartesian plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 4, find the corresponding value of y.

x y

−2 −2

0 2

1 4

3 8

The temperature of a liquid increases as heat is applied. The temperature y (in °C) is related to the time t (in minutes) by the equation y = 3t + 7. a

Complete the table of values: Time (t minutes) Temperature ( y °C)

Plot the points from the table on a Cartesian plane.

Is the relationship linear or non-linear?

Chapter 10 review mathspace.co

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The profile of a mountain trail is on the graph. A hiker starts at point A, navigates the slope through B and reaches ground level at point C. a

When the hiker is at point B: i ii

y 5m A 4m

ow far have they travelled horizontally from H the start point? ow far have they travelled vertically from H the start point?

How far will they have travelled horizontally from point A the next time they are the same height as point B?

3m 2m

1m 1m

C 3m 4m

The points given represent three vertices of a parallelogram. Find the coordinates of the fourth vertex if it is known to be in the second quadrant.

5 4 3 2 1

x 1

2 3 4 5

After each hour, the number of pages a person reads increases according to the table: Hour No. of pages

388

The hiker plans to set up camp 1 metre above the end of the trail. How far above the ground is this?

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 5

2 9

3 13

4 17

Plot the number of pages against each hour on a Cartesian plane.

Is the relationship between the number of hours and the number of pages linear?

If the relationship between the number of hours and the number of pages is linear, write the equation for the relationship where x is the number of hours and y the number of pages.

Mathspace WA Curriculum – Year 7 mathspace.co

Suppose a factory produces n widgets in h hours. The number of widgets produced is always equal to the square of the number of hours that have passed. a

Complete the table of values: Hours (h) Widgets produced (n)

Which of these equations represents the relationship between the number of hours passed, h, and the number of widgets produced, n? A

n=h

n = h2

n = 2h

n=h+2

Plot the points from the table on a Cartesian plane.

Is the relationship between hours passed and widgets produced linear or non-linear?

Olivia is 18 years old and Emily is 12 years old. Let O represent Olivia’s age and E represent Emily’s age. a

Write an equation for O in terms of E.

Identify whether these ordered pairs, in the form (E, O), satisfy the relationship between Olivia and Emily’s ages:

(12, 18)

(34, 28)

iii

(15, 21)

(22, 28)

(13, 19)

vii (35, 42)

(20, 20)

Plot the points from part (b) that satisfy the relationship between their ages on a Cartesian plane.

Did you know?

The Cartesian plane is essential in digital art! By using x- and y-axis coordinates, artists can place, scale, and align every element — from brushstrokes to background textures — with accuracy and balance. Chapter 10 review mathspace.co

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Big ideas Geometry uses standard vocabulary and symbols to communicate facts and relationships about geometric figures.

Curriculum Outcomes • Explore, identify, define, name, label and apply the language, notation and conventions of geometry for points, lines, angles and polygons • Investigate, identify and describe corresponding, alternate and co-interior angles formed when two parallel lines are crossed by a transversal. Use relationships to find unknown angles and explain reasoning • Demonstrate that the interior angle sum of a triangle is 180° • Explore to classify and name triangles according to their side and angle properties. Use the properties to find unknown angles in triangles and explain reasoning • Analyse, represent and interpret problems in real-world situations involving perimeter and area of squares, rectangles, triangles and rectangular composite shapes, parallel lines, properties of triangles, transformations of points, views of rectangular prisms and rectangular composite objects, volume and/or Australian time zones

11 Angles, lines and shapes Chapter outline 11.01 Geometry notation and language 11.02 Measure angles 11.03 Parallel lines and transversals 11.04 Triangles 11.05 Quadrilaterals Investigation: Sort shapes with algorithms Chapter 11 review

392 401 410 422 432 442

The world’s longest stretch of straight railway stretches 478 km across the Nullarbor Plain in Australia.

11.01 Geometry notation and language After this lesson, you will be able to… • identify and name points. • identify and name segments, rays and lines. • name the angles formed by intersecting segments, rays or lines.

Points and segments A point is a single location, with no height or width. We use capital letters to distinguish two different points. This diagram shows three different points labelled A, B and C.

If we connect all of the points from one point to another, we make a segment. This is the segment between the points A and B.

A segment always contains its endpoints. We will sometimes say it is the segment from A to B, which is the same as the segment from B to A. We will often use the abbreviation AB to mean this segment.

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We place small markings on segments when we want to show that they are equal in length. C

AB = AC

In this diagram, the segment AB has the same length as the segment AC.

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This does not mean that the two segments are made up of the same points – only that they have the same length. Sometimes we will use more than one kind of marking to show that some segments are equal to others. B

In this diagram, we use both one-stroke markings and twostroke markings.

C AB = BC CD = DE

A D

Example 1 W

This diagram has two equal segments marked:

Z Y

What segment is equal in length to ZY?

Create a strategy Choose the segment with the same marking as the given segment.

Apply the idea Segment XW has the same marking as ZY. So segment XW is equal in length to ZY.

Idea summary A point is a single location, with no height or width and can be labelled as a single letter like A or B. A segment is a composition of points connected together between two endpoints. B

We place small markings to show the segments that are equal in length.

C AB = BC CD = DE

A E

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Rays If we start at one point and keep going, we make a ray. A

This is the ray from A through B.

And this is the ray from B through A.

Direction is important for rays – these two objects are not the same.

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Idea summary A ray is a composition of points connecting two points that goes through one of those points. The ray extends through the second point in the name of the ray. For instance, for ray AB, the ray extends through B, not A.

Lines If we keep going in both directions, we make a line through A and B:

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To summarise, segments and lines stay the same if we reverse the order of the points, but this is not true for rays: X

line through X and Y = line through Y and X X

segment X Y = segment Y X

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ray from X through Y X

ray from Y through X

Example 2 Select the diagram that shows the line through E and F : A

G F

Create a strategy

Apply the idea

The line should pass through both points and have arrows at both ends.

We need to choose an option where an extending line passes through points E and F. The answer is option C.

Idea summary A line between two points contains every point between the points and all the points beyond on either side. A ray starts at one point and continues through another and beyond. A segment starts at one point and stops at the other. Rays go through different directions when the points are reversed which makes it different from segments and lines.

line ray segment

Angles Whenever two lines, rays or segments pass through the same point, we can describe the relative orientation of one to the other using an angle.

Arm

Here are two rays drawn from the same point through two other points. The rays are also called arms.

Vertex

The point where the arms meet is called the vertex. Arm

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There are two ways to turn from one to the other. The shorter turn is simply called the angle between the objects and the larger turn is called the reﬂex angle. We draw a circular arc from one object to the other to denote the angle (or reﬂex angle).

angle reflex angle

We can use three points to refer to an angle by using the symbol “∠” followed by three letters, one for each point. The ﬁrst letter will be on one of the rays, lines or segments, the second point will be their intersection and the third will be a point on the other ray, line or segment. This means there are two equally valid ways to refer to an angle, as illustrated in this diagram: 1

BAC =

A 2

C 3

BAC

CAB

A 2

C CAB

Just like with segments, we can use additional markings to show that two angles are equal. We draw multiple arcs to show that different angles are equal to each other. B F

In this diagram, the two angles drawn with double arcs are equal. E

C BAC =

DCE D

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Example 3 A

This diagram has the angle ∠ABC marked. What is another way of referring to the same angle? B

Create a strategy

Apply the idea

We can reverse the letters in the original name to ﬁnd a new name for the angle.

Reversing the points in ∠ABC gives us ∠CBA.

Idea summary The angle between two intersecting segments, lines or rays represents their relative orientation to each other. We write the angle symbol “∠” followed by three letters. The second letter is always the intersection point and the ﬁrst and third letters lie on the objects forming the angle, one on each. Swapping the ﬁrst and third letters does not change the angle. An angle has a vertex and two arms.

11.01 Practice questions Understanding 1

Explorer 1–3

Adventurer 4

Trailblazer 4

Identify each as line, ray, or segment: a

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Draw these geometric objects: a

The line through F and G.

The segment BA.

The ray from P through Q.

The ray from Z through Y.

The line through E and F.

The segment EG.

The ray from A through B.

Describe each geometric object using the points A and B: a

A B

Are these statements true or false? a

The order of points matters when naming a line segment.

A ray has a deﬁnite starting point but no endpoint.

The markings on segments indicate their length, but not whether they are equal.

When naming an angle, swapping the ﬁrst and third letters changes the angle.

Fluency Ex 1

5–7, 10

In these diagrams, name the equal sides: a

Ex 2

5, 8–10

Q B

Each diagram shows two lines interesecting at X: a

Describe a line that is the same as the line through X and Q. P

X R Q

Identify a ray that is the same as the ray from C through A.

P X

C B Q

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O R

This diagram shows a line through P, Q and R: Describe a ray that is the same as the ray from P through R.

R Q P

Ex 3

In the given diagram, the angle can be labelled ∠GMT. Write another way of labelling the same angle.

M G

Name the angles that are highlighted in these diagrams: a

D b

D c

D d

Z W

B C

X E

Y X

Name the equal angles in these diagrams: a

A B

C B

B C

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Reasoning

11, 12

If point Q is on line PR, what does that tell us about how point Q and line PR are connected? Explain your reasoning.

Explain how the naming of the rays differs from the naming of line segments and why that is.

For each geometric ﬁgure, determine a real life example which is a model for the ﬁgure. Are there any limitations or restrictions on the real life example? a

A ray

Problem-solving 14 14

A line

15–17

Consider this diagram: If the length of segment AB is 5, ﬁnd the length of segment BC.

In the given line PT, determine how many line segments are there. Make sure that PQ and QP are not counted as separate segments since they represent the same segment.

S R Q P

Determine the number of angles formed within these diagrams: a

Draw the diagram based on these conditions: • Line AB and line CD represent the same line. • Line CB and line CD represent the same ray. • Line CA and line CD represent two rays that go to opposite directions. • CB = CD

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11.02

Measure angles

After this lesson, you will be able to… • identify different types of angles. • understand the relationships between angles, including adjacent angles, vertically opposite angles, complementary angles and supplementary angles.

Angle types An angle is formed between two lines, rays or segments whenever they meet. These lines are called the arms of the angle. The point at the centre of the angle is called the vertex. An important angle in geometry is the angle 90°. We call this special angle a right angle and we draw it with a small square rather than a circular arc. A right angle represents a quarter of a turn around a circle. When lines form a right angle, we say they are perpendicular. A right angle is 90°. The symbol for a perpendicular line is ⊥.

We draw all other angles with a circular arc. An angle that is smaller than a right angle is called an acute angle. An acute angle is an angle that is between 0° and 90°.

An angle that is greater than a right angle but smaller than a straight angle is called an obtuse angle. An obtuse angle is an angle that is between 90° and 180°.

Two right angles together form a straight angle. This is the angle on a straight line. A straight angle is 180°.

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A reﬂex angle is larger than a straight angle, but smaller than a full revolution. A reﬂex angle is an angle that is between 180° and 360°.

Four right angles is the same as two straight angles, making a full revolution. A full revolution is 360°.

Example 1 Select the obtuse angle: A

Create a strategy

Apply the idea

An obtuse angle is greater than 90°, but less than 180°.

The answer is option C.

Idea summary

Right angle

Acute angle

Obtuse angle

Straight angle

Reflex angle

Full revolution

• A right angle is 90°. • An angle that is smaller than a right angle is called an acute angle. • An angle that is greater than a right angle but smaller than a straight angle is called an obtuse angle. • Two right angles together form a straight angle. This is the angle on a straight line. • A reflex angle is larger than a straight angle, but smaller than a full revolution. • Four right angles is the same as two straight angles, making a full revolution.

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Measure angles The units we use to measure angles are called degrees and we write this unit using a small circle like this: 360°. One degree comes from dividing a full revolution up into 360 parts. Since 90 is one quarter of 360, we know that a right angle is exactly 90°. This circle has markings every 45°.

90° 135°

45°

Start

180°

225°

315° 270°

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This lets us associate numbers with the angle types we learned about above. A full revolution is made up of 360°, a single degree is written as 1°. Angle type Acute angle Right angle Obtuse angle Straight angle Reflex angle Full revolution

Angle size Greater than 0°, smaller than 90°. 90° Greater than 90°, smaller than 180°. 180° Greater than 180°, smaller than 360°. 360°

Example 2 Select the angle that is closest to 120°: A

Create a strategy

Apply the idea

120° is greater than a right angle (90°) and smaller than a straight angle (180°).

The answer is option C.

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Idea summary A full revolution is made up of 360 degrees, a single degree is written 1°. Angle type Acute angle Right angle Obtuse angle Straight angle Reflex angle Full revolution

Angle size Greater than 0°, smaller than 90°. 90° Greater than 90°, smaller than 180°. 180° Greater than 180°, smaller than 360°. 360°

Angle relationships Adjacent angles share a common arm and common vertex.

Vertically opposite angles are equal in size.

Complementary angles add to 90° and supplementary angles add to 180°.

404

Complementary

Supplementary

Forming right angle 90°

Forming straight angle 180°

Mathspace WA Curriculum – Year 7 mathspace.co

Example 3 The angles in the diagram are complementary. What is the value of x?

x° 39°

Create a strategy Complementary angles sum to 90°.

Apply the idea x + 39 = 90

Equate the sum of the angles to 90

x = 90 - 39

Subtract 39 from both sides

x = 51

Evaluate

Idea summary Adjacent angles are two angles sharing a common arm and common vertex. Vertically opposite angles are two pairs of equal angles formed whenever two segments, lines or rays intersect at a point. Two angles that form a right angle are complementary and they sum to 90°. Two angles that form a straight line are supplementary and they sum to 180°.

11.02 Practice questions Understanding 1

Explorer 1

Adventurer 1, 2

Trailblazer 2

Which of these diagrams has a pair of adjacent angles shown in red and blue? a

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Are these statements true or false? a

If two angles sum to 90°, they are complimentary.

Vertically opposite angles can be adjacent.

If two angles are supplementary, they sum to 180°.

Two obtuse angles that are adjacent form another obtuse angle.

If two angles form a full revolution, then one of them must be greater than or equal to 180°.

Complementary angles that are adjacent form a straight angle.

Fluency Ex 1

3–6

3–6, 8

Classify each as either a/an: • Reﬂex angle • Straight angle • Revolution

• Right angle • Acute angle • Obtuse angle

3, 4, 6–8

Classify each pair of angles using any of the terms in the list that apply: • Adjacent • Complementary • Supplementary • Vertically opposite angles a

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Ex 2

Approximate these angles in degrees and then label each one as acute, right, obtuse, straight, reflex or revolution. a

For each pair of angles, decide if they are: • Complementary angles • Supplementary angles • Neither a

160°

19°

25° 65°

140° 40°

34° 55°

Given that lines BE and CF are straight lines, use the diagram to identify: a

A pair of adjacent angles.

A pair of vertically opposite angles.

A pair of complementary angles.

D F

In the diagram, ﬁnd: a

An angle that is vertically opposite to ∠CXD.

An angle that is supplementary with ∠AXB.

A P

R C

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Reasoning 9

9, 10

9–11

Decide whether each diagram shows the given angle pair. Explain your reasoning. a

Complementary angles

Vertically opposite angles 28° 28°

68° 12°

Supplementary angles

71° 109°

If ∠ABC and ∠DBF have the same measure, can we say that ∠ABC and ∠DBF are vertically opposite angles? Explain your answer.

Consider the question which was written poorly: Find the measure of ∠Q. S

37° 22° Q

Two students are debating on the answer to the question. Presley thinks ∠Q should be 59°. Qing thinks ∠Q should be 22°. Determine the issue with this question and explain why the two students chose different answers.

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Problem-solving 12a-e, 13ab Ex 3

12, 13a-d

12–13

Find the value of x in each diagram: a

x°

x° 40°

50°

x°

83°

130°

94°

f 150° x°

x°

h 10° x°

40°

x°

Find the value of the variable in each diagram: a

x° x°

x°

y°

y° y°

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d 50°

z° z° z° z°

z°

50°

x°

50°

55°

x°

z° z°

x° x°

x° 160°

x° x°

x°

11.03 Parallel lines and transversals After this lesson, you will be able to… • identify and explain the concept of parallel lines and transversals in geometry. • define and recognise co-interior, alternate and corresponding angles formed by a transversal crossing parallel lines. • apply the knowledge of angle relationships to solve problems in geometry.

Parallel lines and transversals Parallel lines are like train tracks that never meet. They are straight lines that are always the same distance apart and go in the same direction.

Parallel

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Not parallel

Parallel lines are marked with arrows called chevrons. If the number of chevrons on two lines match, they are parallel. The two green lines are parallel so we mark them both with one arrow. The two black lines are parallel so we mark them both with two arrows.

A transversal is a straight line that cuts across two or more other lines.

A B

If two lines are parallel, we can indicate that using the symbol ||. In this case we can say that lines AB and CD are parallel by writing AB || CD.

C D

Example 1 Select the diagram showing a pair of parallel lines: A

Create a strategy

Apply the idea

Look for the pair of lines that have the same number of arrow markings drawn on them to indicate they are parallel.

The lines in option A have the same markings on them showing that the lines are parallel. The answer is option A.

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Idea summary Parallel lines are lines that never intersect. A transversal is a line that cuts across two or more other lines.

Co-interior and alternate angles When a transversal cuts across two lines, three pairs of angles are created. Co-interior angles lie between two lines and on the same side of a transversal. Co-interior angles form a ‘C’ shape with the parallel lines and the transversal. The red and blue angles form a co-interior pair. If the lines are parallel, then the co-interior angles are supplementary (add up to 180°).

Alternate angles form a ‘Z’ shape with the parallel lines and the transversal. If the lines are parallel, then alternate angles are equal in size.

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Example 2 Select the diagram showing a pair of alternate angles: A

Create a strategy Choose the option that shows two angles formed by the parallel lines and are located on opposite sides of the transversal.

Apply the idea In option D, both angles are between the parallel lines and are on opposite sides of the transversal. The correct answer is option D.

Example 3 Solve for the value of x.

x° 128°

Create a strategy Identify the relationship between the red and green angles. The angles lie between the two lines on the same side of the transversal so are a pair of co-interior angles.

Apply the idea The angles are co-interior angles which are supplementary. x + 128 = 180 Equate the sum of the angles to 180 x = 180 − 128 Subtract 128 from both sides x = 52

Evaluate

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Idea summary When a transversal crosses a pair of parallel lines: • Co-interior angles are supplementary (add to 180°). • Alternate angles are equal in size.

Corresponding angles Corresponding angles lie on the same side of the transversal and both above or both below the parallel lines. Corresponding angles form an ‘F’ shape with the parallel lines and the transversal. If the lines are parallel, the corresponding angles will be equal.

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Example 4 Select the diagram showing a pair of corresponding angles: A

Create a strategy

Apply the idea

Choose the option that shows two angles formed on the same side of the transversal and either above or below the parallel lines.

If we look at option C, we will see that both angles are on the right side of the transversal and both below the parallel lines. Therefore, the correct answer is option C.

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Example 5 Solve for the value of x. 125° x°

Create a strategy Identify the relationship between the angles shown. The angles are on the same side of the transversal and both below the pair of lines so are a pair of corresponding angles.

Apply the idea The angles are corresponding angles, which are equal. x = 125 Equate the corresponding angles

Idea summary Corresponding angles lie on the same side of the transversal line and both above or both below the pair of lines. If the pair of lines are parallel then the corresponding angles are equal in size.

11.03 Practice questions Understanding 1

Explorer 1, 2

Adventurer 1, 2

Trailblazer 2, 3

How do we indicate that lines are parallel on a diagram?

Draw an example of correctly marked parallel lines.

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These corresponding angles can be thought of as forming an ‘F’ shape:

What shape do these co-interior b angles make?

What shape do these alternate angles make?

Are these types of angles equal, complementary (add to 90°) or supplementary (add to 180°)? a

Alternate angles

Fluency Ex 1

4–11

Corresponding angles

5–11, 13, 14

8–14

Identify the type of angle relationship shown: a

Ex 3

Co-interior angles

Identify whether the pairs of lines shown are parallel or non-parallel: a

Ex 2

For each diagram, find the value of x and justify your answer. a

c x°

x° 64°

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107°

x° 96°

Ex 4

Identify the type of angle relationship shown:

For each diagram, name the angle which is: i

Alternate to the marked angle

Co-interior with the marked angle

iii

Corresponding to the marked angle

P Q

C F

Line AE is parallel to BD Line BF is parallel to CE Ex 5

For each diagram, find the value of x and justify your answer. a

b 110°

x° x°

x°

75°

125°

The lines GH and KL are parallel. a Which angle corresponds to angle 1? b Which angle is co-interior with angle 6? c

Which angle is alternate to angle 5?

Which angle(s) are supplementary to angle 3?

If angle 7 is 105°, what is the size of angle 4?

If angle 4 is 95°, what is the size of angle 6?

If angle 3 is 40°, what is the size of angle 5?

2 4

1 3 6

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The lines A and B are parallel and lines C and D are parallel. a Name the angle alternate to angle 1. b Name the angle corresponding to angle 4. c

If angle 2 is 56°, what is the size of angle 3?

If angle 4 is 113°, what is the size of angle 1?

2 1

3 4

The lines A, B and C are parallel. a Name all the angles that are equal in size to angle 1. b

Name all the angles that are equal in size to angle 4.

Name four pairs of vertically opposite angles.

If angle 4 is 112°, what is the size of angle 8?

If angle 5 is 105°, what is the size of angle 9?

Determine the size of angle 6.

1 5 6 9 13

∠NBP and ∠DCR

∠BCQ and ∠TDA

∠NBA and ∠LAD

∠MAL and ∠TDS

3 4 7

A 8

10 14

Which of these pairs of angles are equal? If they are, justify by naming the angle relationships. A

N M

B L

C D

T S

Which of these pairs of angles are equal? If they are, justify by naming the angle relationships. A

∠NBA and ∠QCB

∠BAD and ∠ADC

∠ADC and ∠DCR

∠ABC and ∠BAD

N M

B L

C D

T S

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Reasoning 15

15bdf, 16

15bdf, 16, 17

For each diagram, state whether there are parallel lines. Explain your answer. a

b 120° 50°

70° 110°

d 79°

77°

74° 77°

102°

102° 102°

112°

78°

73°

53° 127°

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Is there a pair of parallel lines in this diagram? Explain your answer.

a b

c d

Consider the given diagram: a

Solve for p.

Justify your answer to part (a).

Solve for q.

Justify your answer to part (c). 26° p°

Problem-solving 18–20 18

18, 20–23

q°

21–26

For this diagram: a

How are ∠BPD and ∠AQE related?

How are ∠AQE and ∠DQE related?

If ∠BPD = 84°, ﬁnd the value of ∠DQE.

Q P

D E

For this diagram: a

How are ∠APB and ∠CPE related?

How are ∠CPE and ∠BQD related?

If ∠APB = 107°, ﬁnd the value of ∠BQD.

A D F

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Q E

For this diagram: a

How are ∠DQF and ∠CPE related?

How are ∠CPE and ∠EPF related?

If ∠DQF = 97°, ﬁnd the value of ∠EPF.

A F

Q C D

Solve for the value of x in the diagram shown.

(3x − 4)° 113°

Solve for the values of x and y. a

(y + 12)°

x°

114°

b (3x − 15)°

(y + 25)°

105°

For this diagram: a

Find the value of x.

Find the value of y.

63°

y° x°

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For this diagram: a

Find the value of x.

Find the value of y.

Find the sum of the 3 interior angles of the triangle. 33°

y°

For this parallelogram: a

Find the value of x.

Find the value of y.

x°

(5x + 3)°

148°

(6y)°

Find the values of x, y and z.

x° 133° y°

53°

47°

z°

11.04

Triangles

After this lesson, you will be able to… • classify triangles based on their properties. • apply angle sum and exterior angle properties of triangles to solve problems.

Classifications of triangles A triangle is a 2D shape with three straight sides and three angles. Triangles can be classiﬁed according to their interior angles or sides. Classiﬁcation by interior angles: • If all the angles are acute, the triangle is an acute triangle. • If one of the angles is a right angle, the triangle is a right-angled triangle. • If one of the angles is obtuse, the triangle is an obtuse triangle.

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Classiﬁcation by sides: • If all the sides have different lengths, the triangle is a scalene triangle. • If two sides have the same length, the triangle is an isosceles triangle. • If the three sides have the same length, the triangle is an equilateral triangle. Scalene All sides different

Isosceles Two sides same

Equilateral Three sides same

Acute All angles acute

Right One right angle

Obtuse One obtuse angle

Isosceles triangles have a special property. The angles opposite the equal sides are also equal. In this triangle, x = 70. 6

x°

70°

Equilateral triangles have all angles equal to 60°. 60°

60°

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Example 1 Consider the triangle shown. a Which of the following words describes this triangle? A

Scalene

Equilateral

Isosceles

Create a strategy

Apply the idea

Take note of the sides marked as equal.

The answer is option C: isosceles, because two sides of the triangle were marked equal.

b Which of the following words also describes this triangle? A

acute

right-angled

obtuse

Create a strategy Consider the size of the interior angles of the triangle. • If all the angles are acute, the triangle is an acute triangle. • If one of the angles is a right angle, the triangle is a right-angled triangle. • If one of the angles is obtuse, the triangle is an obtuse triangle.

Apply the idea All the angles of the triangle are acute angles. So the correct answer is A: acute.

Idea summary Triangles can be classiﬁed by their angles and their sides as shown in the diagram: Scalene All sides different

Acute All angles acute

Right One right angle

Obtuse One obtuse angle

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Isosceles Two sides same

Equilateral Three sides same

Angle sum of a triangle Interactive exploration

Discover this concept in action online

65°

50°

mathspace.co

The sum of the interior angles of a triangle is 180°. 65° + 65° + 50° = 180°

65°

Example 2 Consider the triangle shown. 49°

41°

Is it a right-angled triangle? A Yes

Create a strategy We need to ﬁnd the remaining angle. Let x be the remaining angle, add it to the given angles and equate the sum to 180°.

Apply the idea x + 49 + 41 = 180 Add the angles and equate to 180 x + 90 = 180

Evaluate the addition

x + 90 − 90 = 180 – 90 Subtract 90 from both sides x = 90°

Evaluate

Yes, it is a right-angled triangle. So the correct answer is A.

Idea summary The sum of the angles in a triangle is 180°.

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Exterior angles in a triangle The angle formed outside the triangle by extending a side is called an exterior angle.

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Here the exterior angle is ∠PAB. The size of an exterior angle is always equal to the sum of the interior angles on the opposite side. B

∠PAB = ∠ABC + ∠ACB We know that the sum of the interior angles is 180°. So, 180 = ∠CAB + ∠ABC + ∠ACB We also know that angles on a straight line sum to 180°. So, 180 = ∠PAB + ∠CAB Since both sums are equal to 180°, we can equate them: ∠PAB + ∠CAB = ∠CAB + ∠ABC + ∠BCA ∠PAB = ∠ABC + ∠ACB

Example 3 Solve for the value of x in the diagram shown. 52° x° 108°

Create a strategy Use the exterior angle relationship.

Apply the idea x + 52 = 108 x + 52 − 52 = 108 – 52 x = 56

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Add the interior angles and equate to 108 Subtract 52 from both sides Evaluate

Idea summary An exterior angle is equal to the sum of the opposite and non-adjacent interior angles.

11.04 Practice questions Understanding 1 2

Adventurer

Trailblazer

2, 3

2–4

Fill in the blank: The sum of the interior angles of any triangle is ⬚.

Describe the features of a triangle that classify it as: a

Explorer 1–3

Acute

Obtuse

Right-angled

Describe the features of a triangle that classify it as: a

Scalene

Identify 3 different exterior angle relationships.

How are angles BDE and CDF related?

Isosceles

Equilateral A

D F

Fluency Ex 1

5a-d, 6a-d, 7a-c, 8–9, 10a-c

Classify each triangle by choosing one classiﬁcation from list 1 and one classiﬁcation from list 2:

5–9, 10d-f

List 1 Acute Obtuse Right-angled

g 66° 24°

5–9, 10d-f

List 2 Isosceles triangle Equilateral triangle Scalene triangle d

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Decide whether each triangle is scalene, isosceles or equilateral: a

d 8

g 5

8 7

7 Ex 2

Decide whether these triangles are acute, obtuse or right-angled: b

38°

41°

47°

62°

43°

44°

122°

41°

19°

98°

Find the measure of the third angle of a triangle given the measures of the other two angles are: a

36° and 94°

45° and 45°

3° and 150°

lassify the triangle by its interior angles C and sides.

For each diagram: i

Find the value of x.

x° 30° 65° x° 40°

45°

Ex 3

Find the value of x. a

b 30°

x°

110° 55° 70°

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x°

d x°

38° x°

34°

73°

54° 72°

39°

x°

Reasoning 11

73°

11, 12

11, 14

x°

13–15

In this diagram, one way to ﬁnd the value of x is by using the exterior angle theorem: 52°

x + 52 = 108

x° 108°

Explain a different way to ﬁnd the value of x.

Explain why it is not possible to make this triangle. 98°

65° 33°

Miriam has an idea that she shares with her mathematics class. Miriam’s claim is that each pair of interior angles in a triangle has a sum that is equal to two different exterior angles. Is this correct? Provide a diagram to support your answer.

A triangle has two angles with measures of 37° and 85°. Find the measure of the smallest exterior angle of the triangle. Explain your reasoning.

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Determine whether each statement is true or false. If false, provide an example to show it is false, then correct the statement to make it true. a

Only the interior angles of right triangles sum to 180°.

The measure of an exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.

The sum of the measures of the exterior angles of a triangle is 180°.

In a right-angled triangle, the right angle is 90° and the remaining two angles are always equal.

Problem-solving 16, 17ab, 18ac 16

16, 17a-d, 18a-c, 19–21

16–18, 20, 21

Find the value of x. a

x°

135°

b x°

112°

48°

x°

110°

82°

68°

x° x°

For these diagrams: a

Find the values of x and y.

Find the values of a and b.

52°

x° 63°

54°

Find the values of x and m.

Find the values of a and b. 47°

58° 19° x°

x°

b°

m°

a°

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a°

61°

b°

y°

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a°

Find the values of x and y. a

x°

84° y°

65°

72°

44°

65° y° x°

56°

y° 77°

33°

120° x° y° 55°

While camping, Jena was told to have her supporting ropes at an angle of 25°. A picture of her tent is shown.

She knows the tent is on level ground so ∠BCA = 90° and ∠ABC = 65°. Determine whether Jena has pitched her tent correctly, so the supporting ropes are at an angle of 25°. A

For this diagram: a

Identify the relationship between ∠JQR and ∠JNP.

Identify the relationship between ∠MNK and ∠JNP.

Use your answers from parts (a) and (b) to ﬁnd the value of x.

Q J

K L 156°

x° N

115° R

For this diagram: a

Identify the relationship between ∠JHM and ∠KMH.

Use your answer from part (a) to ﬁnd the value of x.

J 68°

92° P x° K

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11.05 Quadrilaterals After this lesson, you will be able to… • classify quadrilaterals as convex or concave based on their internal angles. • identify special quadrilaterals. • describe the properties of different types of quadrilaterals.

Quadrilaterals A quadrilateral is a 2D shape that has four vertices (corners), four sides and four internal angles.

Quadrilaterals can always be split into 2 triangles by connecting a pair of opposite vertices. Since the angle sum of a triangle is 180°, the angle sum of a quadrilateral is the angle sum of two triangles or 360°. Quadrilateral

Split into two triangles

Quadrilaterals can be classiﬁed as convex or concave: Convex

All angles less than 180°. All four vertices point outwards.

Convex

Concave

One internal angle greater than 180°. One vertex points inwards.

Concave

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Example 1 Solve for the value of x in the diagram shown.

64° x°

19°

23°

Create a strategy Add the interior angles and equate to 360.

Apply the idea x + 64 + 23 + 19 = 360 Add the angles and equate to 360 x + 106 = 360 Evaluate the addition x + 106 − 106 = 360 – 106

Subtract 106 from both sides

x = 254 Evaluate

Idea summary A quadrilateral is a 2D shape that has four sides. Its internal angles sum to 360°. A convex quadrilateral is a shape where all the vertices point outwards. All of the internal angles are less than 180°. A concave quadrilateral is a shape where one of the angles is reflex, larger than 180°. One vertex points inwards.

Special quadrilaterals A parallelogram is a quadrilateral with two pairs of parallel sides.

Parallelograms always have all of these properties: • Opposite sides are parallel • Opposite sides are equal in length • Opposite angles are equal • Consecutive angles (angles that share a side) are supplementary (add to 180°) 11.05 Quadrilaterals mathspace.co

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A rhombus is a parallelogram which has all sides the same length: Properties of rhombuses: • Same as parallelogram, plus • All sides are equal in length

A rectangle is a parallelogram with four right angles. Properties of rectangles: • Same as parallelogram, plus • All angles are right angles • Opposite sides are parallel and equal in length A square is a rectangle that has all four sides the same length. Properties of squares: • Same as parallelogram, plus • All sides are equal in length • All angles are right angles A trapezium has two parallel sides. Properties of trapeziums: • One pair of parallel sides • The co-interior angles between the parallel lines are supplementary

A kite has no pairs of parallel sides. Properties of kites: • Two pairs of equal adjacent sides • Two equal angles

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Example 2 Select all the parallelograms: A

Create a strategy

Apply the idea

Recall that any shape that has two pairs of parallel sides is a parallelogram.

The answers are options B and C.

Example 3 Does the quadrilateral shown qualify as a trapezium?

113°

67°

Create a strategy Add the angles to check if the sum is 180°.

Apply the idea Sum of angles = 67° + 113° = 180°

Add the angles Evaluate

Since the sum of the angles that are on the same side is 180°, they are supplementary, which makes the opposite sides parallel. So, the quadrilateral is a trapezium.

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Idea summary A parallelogram has two pairs of parallel sides and opposite angles are equal. A rhombus is a parallelogram with four equal sides. A rectangle is a parallelogram with four right angles. A square is a rectangle with four equal sides. A trapezium has one pair of parallel sides and the co-interior angles between the parallel sides are supplementary. A kite has two pairs of equal adjacent sides and one pair of equal opposite angles. The properties are summarised in this diagram: Trapezium One pair of parallel sides

Parallelogram Two pairs of parallel sides

Rhombus All sides equal

Kite Two pairs of adjacent equal sides

(Opposite sides are equal) Rectangle All angles right

Square

11.05 Practice questions Understanding 1

436

Explorer 2–4

Adventurer 1, 3, 4

Are these statements true or false? a

All quadrilaterals are parallelograms.

All rectangles are parallelograms.

All parallelograms are rhombuses.

A parallelogram with four equal sides is a square.

A rectangle with two equal connected sides is a square.

A rhombus is a parallelogram.

All quadrilaterals have an angle sum of 360°.

A quadrilateral can have exactly 3 right angles.

Mathspace WA Curriculum – Year 7 mathspace.co

Trailblazer 1, 3, 4

Can a quadrilateral have exactly 2 right angles? If so, name a quadrilateral that could have two right-angles.

In the quadrilateral ABCD, side AB has a length of 6 units. State the other side of ABCD which must have a length of 6 units.

D C

State whether each quadrilateral is convex or concave: a

5–8, 10, 11

5–11

Fluency Ex 1

5a-d, 6a-d, 7, 11a-c

Find the value of x. a

120°

60°

80°

60°

95° x°

155°

x° 35°

70°

x°

46°

123°

74°

112° 131°

63°

x°

68° 136°

35°

x°

108° 107°

83°

57°

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Ex 2

Ex 3

Classify each quadrilateral using all the terms that apply out of: rectangle, rhombus, kite, parallelogram, square, trapezium. a

Decide whether each quadrilateral is a trapezium, parallelogram or neither: a

59° 97° 73°

121°

d 66°

66°

114°

116°

114° 64°

66°

438

Sketch each parallelogram, labelling all sides and angles accordingly: a

A parallelogram that is neither a rectangle nor a rhombus.

A parallelogram with a pair of connected sides that are equal and a pair of acute angles.

A parallelogram with a right angle.

Mathspace WA Curriculum – Year 7 mathspace.co

Which two sides are parallel in this trapezium?

B 99°

81°

80°

100°

Find the values of angles a, b and c. a

R Q c

a a

77°

111°

S J

c b M

B C a

b Y

60°

b a

D 105° X

c W

For each quadrilateral: a

Find the values of x and y. x°

Find the values of a, b and c. a°

y°

117°

73°

81°

c°

b°

Find the values of a and b. a°

Find the values of p and q. 6

b° p

101° q−3

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Reasoning

13, 14, 16, 17

13–16

12–15

What is the greatest number of internal reﬂex angles that a quadrilateral can have? Explain your answer.

What is the greatest number of internal obtuse angles that a quadrilateral can have? Explain your answer.

What is the greatest number of internal right angles that a quadrilateral can have? Explain your answer.

Sheldon draws a quadrilateral and covers it up. He tells Wanda that the quadrilateral has opposite sides that are equal in length. From this information, determine the most specific classification that Wanda can determine for the figure. Explain your answer.

Classify each quadrilateral. Explain your reasoning. a

91°

15 8

97°

83°

97°

8 15

Explain why some quadrilaterals may be classiﬁed as more than one type of quadrilateral.

Problem-solving 18bc 18

18a-c, 19

18–20

Find the value of x. a

(3x + 2)° 114°

b 55°

128°

x°

97°

64° x°

26°

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P 51° (28x − 11)° R

∠CDE is 24°. Populate the rectangle ABCD with all the remaining angle values stating reasoning.

A E

Find the value of x if the perimeter of this parallelogram is 48 m. 8 8x

Investigation: Sort shapes with algorithms

Investigate online

mathspace.co

Did you know?

Origami is built on geometry! Every fold — from sharp mountain creases to delicate valleys — follows precise angles and lines to transform a flat sheet into a stunning three-dimensional work of art.

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11 Chapter review 1

Draw each geometric object: a

The line through P and Q.

The ray from F through G.

The segment XY.

Each diagram shows two lines intersecting at X: a

Describe a line that is the same as the line through X and D.

Identify a ray that is the same as the ray from Q through X.

D A

O P

A X

B Q

Name the equal angles in each diagram: a

L M

C B

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A D

Identify whether the pairs of lines shown are parallel or non-parallel: a

Are these statements true or false? a

All quadrilaterals are rectangles.

All parallelograms are rectangles.

All rhombuses are parallelograms.

A parallelogram with four right angles is a square.

All quadrilaterals have an interior angle sum of 180°.

For the marked angle in each diagram, name the angle which is: i

Alternate

Co-interior

iii b

Q A

D F

Corresponding

Line AC is parallel to FD Line BF is parallel to CE 7

Find the value of x. a

c x°

125° 74°

x°

102°

Solve for x in the given diagram. 3x + 10 124°

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For this diagram: a

Find the value of x.

Find the value of y.

34° y° x°

For this parallelogram: a

Find the value of x.

Find the value of y.

(8y + 2)°

(2x)° 42°

Classify each triangle by choosing one classification from list 1 and one classification from list 2: List 1 Acute Obtuse Right-angled

List 2 Isosceles triangle Equilateral triangle Scalene triangle

f 56°

34°

8 6

5 5

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Find the value of x. a

c x°

40°

x°

65° 110° 53°

x°

45°

e 45° 83°

42°

x°

68°

Find the value of the missing angle marked with x. a

80°

60°

x° 155°

x° 45°

45°

30°

x°

86°

129° 64°

109° x°

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For each quadrilateral: a

Find the values of x and y.

Find the values of a, b and c.

99°

a°

107°

x°

119°

y°

c°

b°

Find the value of x in the diagram.

J 68°

92°

x° K

For these diagrams: a

Find the values of x and y.

Find the values of x and m.

51°

58° 18°

x° 64°

m°

x°

y°

56°

Find the values of x and y. a

x° 83°

b 115°

y°

x° 79°

41°

y° 56°

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Find the value of x. a

b 62°

57° x°

x°

97° 24°

Find the value of x if the perimeter of this parallelogram is 58 m. 5 7x

Did you know?

Parallel lines, like train tracks, never meet, no matter how far they extend? In urban planning, city streets and railway tracks are often designed with parallel lines to ensure straight and efficient routes. These lines help ensure that vehicles, trains or buses can move without crossing paths. Even if you stand at the end of a long highway, the lines might appear to converge in the distance, but they never truly meet.

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Big ideas The formula for the area of a triangle is derived from the area of a rectangle, then applied to find the area of composite shapes and surface area.

Curriculum Outcomes • Establish and apply relationships between lengths of sides, perimeter and area for squares, rectangles and triangles. Generalise and apply formulas, using appropriate units • Explore and explain efficient strategies to determine the perimeter and area of irregular or composite shapes composed of squares and rectangles • Explore and establish connections and conversions between units of area • Analyse, represent and interpret problems in real-world situations involving perimeter and area of squares, rectangles, triangles and rectangular composite shapes, parallel lines, properties of triangles, transformations of points, views of rectangular prisms and rectangular composite objects, volume and/or Australian time zones

12 Length and area Chapter outline 12.01 12.02 12.03 12.04 12.05 12.06 12.07

Units of length Perimeter Perimeter of composite shapes Units of area Area of rectangles and squares Area of triangles Area of composite shapes Modelling investigation: Solar panels Chapter 12 review

450 456 466 472 480 489 496 506 509

The distance a bicycle travels in one spin of the wheel is equal to the circumference of the wheel. We can calculate that using π and the diameter of the wheel!

12.01

Units of length

After this lesson, you will be able to… • recognise the common units of length, including millimetres, centimetres, metres and kilometres. • convert lengths between units of length. • compare different lengths by converting them to the same units. • perform two-step conversions.

Conversion factors The most common units of length we will encounter are: • millimetres (mm): a grain of sand has a length of around 1 mm. • centimetres (cm): the length of an ant is around 1 cm. • metres (m): the length of a desk is around 1 to 2 m. • kilometres (km): the Sydney Harbour Bridge is just over 1 km long. To convert from one unit to another, we will make use of these relationships: 1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm The conversion factors are summarised in the conversion chart below: ÷ 10

÷ 100

× 10

÷ 1000

× 100

Example 1 Convert 6.52 centimetres to millimetres.

Create a strategy There are 10 mm in 1 cm so to convert from cm to mm use ×10.

Apply the idea 6.52 cm = 6.52 × 10 = 65.2 mm

450

Multiply 6.52 by 10 Evaluate

Mathspace WA Curriculum – Year 7 mathspace.co

× 1000

Example 2 Victoria is 1.14 m tall and Joanne is 156 cm tall. Find the taller person by converting both lengths to the same unit of measurement. a Convert both heights to the same unit.

Create a strategy There are 100 cm in 1 m so to convert from m to cm use ×100.

Apply the idea Height = 1.14 × 100 = 114 cm

Reflect and check Multiply 1.14 by 100 Evaluate

We could have converted both heights to the larger unit of metres. Victoria is 1.14 m and Joanne is 1.56 m.

b Who is taller?

Create a strategy Use the result found from part (a) and compare the values.

Apply the idea

Reflect and check

Since 114 cm < 156 cm, Joanne is taller than Victoria.

If we had converted to metres, then when comparing, we would see that 1.14 m < 1.56 m. It doesn’t matter which way you convert, as long as you get a common unit before comparing.

Idea summary To convert from one unit to another, we use the relationships pictured here: ÷ 10

÷ 100

× 10

÷ 1000

× 100

× 1000

To compare different lengths, ﬁrst convert to the same units of length.

Two-step conversions Sometimes we need to convert from one unit to a much larger or much smaller unit. Find the starting unit and end unit on the conversion chart to determine how many conversion steps you will need to do. 12.01 Units of length mathspace.co

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Example 3 Convert 14 metres to millimetres.

Create a strategy

Apply the idea

Start at m on the conversion chart and see that To convert 14 m to mm, to get to mm we need to do two steps: ×100 and 14 m = 14 × 100 × 10 then ×10. = 14 000 mm

Multiply by 100 then by 10 Evaluate

Example 4 Convert 512 200 cm to km.

Create a strategy Start at cm on the conversion chart and see that we need to use two steps to get to km: ÷100 and then ÷1 000.

Apply the idea To convert 512 200 cm into km, 512 200 cm = 512 200 ÷ 100 ÷ 1000

Divide by 100 then by 1000

= 5.122 km

Evaluate

Idea summary To convert with more than one step, first find the starting unit and end unit on the conversion table to find the multiple conversion steps required. ÷ 10

× 10

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÷ 100

÷ 1000

× 100

× 1000

12.01 Practice questions Understanding 1

a e

The length of a bus = 12 ⬚

The width of a shoe box = 10 ⬚

⬚ m = ⬚ km

⬚ cm = ⬚ m

⬚ cm = ⬚ mm

⬚ mm = ⬚ m

When converting from a small unit of measurement (such as m) to a large unit of measurement (such as km), we ⬚ by the conversion factor.

Divide

When converting from a large unit of measurement (such as km) to a small unit of measurement (such as mm), we ⬚ by the conversion factor.

Complete:

d e f

To convert from m to cm, you multiply by ⬚.

To convert from cm to mm, you multiply by ⬚. To convert from km to m, you multiply by ⬚. To convert from mm to m, you divide by ⬚. To convert from cm to m, you divide by ⬚.

To convert from mm to cm, you divide by ⬚.

Fluency

5a-f, 7–9, 11, 14

5, 6a-f, 10, 13–14

5g-l, 6g-l, 10c, 12

Convert: a c e g i k

The height of a chair = 1.2 ⬚

Ex 3, 4

The distance between two cities = 13 ⬚

The length of an ant = 3.7 ⬚

Multiply

Ex 1

The thickness of a book = 2.5 ⬚

Fill in the correct answer to make these statements true:

Trailblazer 3–4

Fill in the boxes to show the relationships between these pairs of units. a

Adventurer 1, 3–4

Identify the appropriate unit of measurement to use for each measurement: c

Explorer 1–4

3 cm to millimetres 6 m to centimetres 6 km to metres 88.2 mm to centimetres 14.9 cm to metres 4.6 cm to millimetres

b d f h j l

60 mm to centimetres 800 cm to metres 9000 m to kilometres 7.33 m to centimetres 12 226 m to kilometres 5.69 km to metres

b d f h j l

3484 mm to metres 866 800 cm to kilometres 1 528 400 mm to kilometres 150 mm to metres 45 300 cm to kilometres 320 000 mm to kilometres

Convert: a c e g i k

24 m to millimetres 6 km to centimetres 9 km to millimetres 9.03 m to millimetres 7.43 km to centimetres 6.71 km to millimetres

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The Qinghai-Tibet railway in China includes the Tanggula railway station. At 5068 metres high, it is the world’s highest railway station. How many kilometres high is the station?

A city is 125 metres above sea level. Shifts in the Earth’s plates make it useful to know the height of the city above sea level in centimetres. Find the height of the city in centimetres.

Ex 2

Purnjay is 176 cm tall. Nadia is 1.51 m tall. Find the shorter person by converting both lengths to the same unit of measurement.

Find the taller person in these pairs: a

Haonan is 165 cm tall and Tina is 1.53 m tall.

Hermione is 1.23 m tall and Sandy is 161 cm tall.

Joanne is 1270 mm tall and Ellie is 1.41 m tall.

A coffee table is 1.17 m long. A whiteboard is 147 cm long. Find the longer item by converting both measurements to the same unit.

Plant 1 grows 3.6 cm in a week. Plant 2 grows 29 mm in the same week. Which plant grew more during the week?

The table shows the different skyscrapers around the world and their heights: Name One World Trade Center Shanghai Tower Petronas Twin Towers Zifeng Tower Burj Khalifa

Height (metres) 541.3

Height (kilometres) 0.632

451.9 0.45 828

Complete the conversions for the given skyscrapers in the table.

Which skyscraper is the tallest?

A desk is 1.39 m long. A rug is 120 cm long. a

Determine whether the desk or the rug is longer.

Calculate the difference between the lengths in centimetres.

Reasoning

15–16

15–17

When you are building a house, all house plan dimensions and building materials are measured in mm. Give a reason why this is the case.

Australia oﬃcially changed all of its units of measurement to the Decimal system in 1974. Prior to this, the Imperial system was used, which included conversions such as 1 foot = 12 inches and 1 mile = 5280 feet. Why do you think the Australian Government made the change to the Decimal system?

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hy can’t metric units such as km be used as a reliable measure of distance in space W exploration and astronomy?

What unit of measurement is used to measure the distance between galaxies and how is it calculated?

Problem-solving 20–22, 26

18, 23–25, 27, 29

19, 23–25, 27–30

Arrange these in descending order of length: 5.23 m, 499 cm, 495 000 mm.

Arrange these in ascending order of length: 0.19 km, 19 150 cm, 192 m.

A rectangular table is measured to be 586 centimetres long and 3.64 metres wide. Calculate the difference between the length and the width in metres.

A car is 4 m long. A truck is 934 cm long. How many metres longer is the truck than the car?

A sail boat is 6 m long. A yacht is 956 cm long. How many centimetres longer is the yacht than the sail boat?

A rectangular table is measured to be 5.23 metres long and 499 centimetres wide. Calculate the difference in metres between the length and the width of the table.

If a ﬂy is 1.16 cm long and a bee is 19 mm long, how much longer is the bee than the ﬂy in centimetres?

If a mosquito is 1.23 cm long and a wasp is 13 mm long, how much longer is the wasp than the mosquito in millimetres?

Michael wants to lay a row of square pavers along the edge of his 3 m garden bed. Find the side length of each paver in centimetres.

Paver Paver Paver

Garden

Paver Paver

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The wingspan of a jumbo jet is 6000 cm:

If 21 jumbo jets are lined up wing-to-wing, ﬁnd the length of the total wingspan in centimetres.

Convert this length to kilometres. 6000 cm

A 100 m support of a bridge is built over a river. Due to water erosion, the riverbed is being lowered by 150 mm every year. In how many years will the bridge collapse due to the decrease in the height of the riverbed?

A giant sequoia tree in a national park is growing at a rate of 500 millimetres per year. The park management is concerned that the tree may fall on a nearby trail that is 10 metres away. How long will it take before the tree poses a threat to the trail?

A rectangular pool measures 10 m by 6 m. If a circular ﬂoat with a diameter of 200 cm is placed in the middle of the pool, calculate the distance of the ﬂoat from both of the edges of the pool.

12.02

Perimeter

After this lesson, you will be able to… • find the perimeter of rectangles, squares and regular polygons. • apply the formulas learned to solve problems involving perimeters. • identify the side length of a square given its perimeter. • identify the length of one side of a regular polygon given its perimeter and number of sides.

Perimeter of a rectangle The perimeter of a shape is the distance around the outside, as perimeter is a length it will always have a unit associated with it. It might be centimetres (cm), metres (m) or it may be generic like using the word units. Every rectangle has two pairs of equal sides, which we can call the length and the width.

Length

Width

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As we can see from the image, the perimeter of a rectangle will always be: Perimeter = Length + Width + Length + Width This is the same as two groups of (Length + Width) which we can write as: Perimeter = 2 × (Length + Width)

Example 1 Find the perimeter of this rectangle: 6m

16 m

Create a strategy Add up the side lengths around the outside of the shape to ﬁnd perimeter.

Apply the idea The given rectangle has a length of 16 m and a width of 6 m. Perimeter = 16 + 6 + 16 + 6 = 44 m Or Perimeter = 2 × (Length + Width)

Add up the sides or use the formula

= 2 × (16 + 6)

Substitute the length and width

= 2 × 22

Add the length and width

= 44 m

Double 22

Idea summary Perimeter is the distance around the outside of a shape. To ﬁnd the perimeter of a rectangle you add up the 4 side lengths. You can also use the formula: Perimeter = 2 × (Length + Width)

Perimeter of a square A square has four equal sides. Therefore the perimeter of a square will be: Perimeter = Length + Length + Length + Length

Side length

This is the same as four groups of the Length, which we can write as: Perimeter = 4 × Length

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Example 2 Find the side length of this square:

P = 32 m

Create a strategy Divide the perimeter by 4.

Apply the idea Side length = 32 ÷ 4 =8m

Divide by 4 Evaluate

Reﬂect and check We could also use the formula for the perimeter of a square: Perimeter = 4 × Length 32 = 4 × Length

Write the formula Substitute the perimeter

Length = 32 ÷ 4

Divide 32 by 4

Length = 8 m

Evaluate

Idea summary To find the perimeter of a square you add up the four side lengths on the outside of the square. You can also use the formula: Perimeter = 4 × Length

Perimeter of regular polygons Regular polygons have all of their sides the same length. Consider a regular hexagon: Since it has six sides of equal length we can write its perimeter as: Perimeter = Length + Length + Length + Length + Length + Length Or more simply: Perimeter = 6 × Length In fact, we can do the same thing for any regular polygon. 458

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Side length

Example 3 Find the perimeter of this shape:

5 mm

Create a strategy Multiply the length of one side by the total number of sides.

Apply the idea We are given a side length of 5 mm. The shape shown is a regular decagon, which means it has 10 sides of equal length. Perimeter = 5 × 10 = 50 mm

Multiply the length by 10 Evaluate

Example 4 This shape has a perimeter of 64 cm. What is the length of each side? P = 64 cm

Create a strategy

Apply the idea

Divide the perimeter by the number of sides. Use the fact that the shape shown is a regular octagon which has 8 sides of equal length.

64 = 8 × Length

Substitute the perimeter

Length = 64 ÷ 8

Divide perimeter by 8

Length = 8 cm

Evaluate

Idea summary To ﬁnd the perimeter of a regular polygon with n sides you multiply the length of one side by n. You can also use the formula: Perimeter = n × Length of one side

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12.02 Practice questions Understanding 1

Trailblazer 2bd

Find the perimeter of these rectangles, where each square is 1 unit. a

Complete the steps to ﬁnd the perimeter of this triangle: P = 55 + ⬚ + ⬚ = ⬚ cm

460

Adventurer 1, 2bc, 3

In these diagrams, each grid square represents 1 square unit. Find the perimeter of each shape: a

Explorer 1, 2ac, 3

Mathspace WA Curriculum – Year 7 mathspace.co

55 cm 48 cm 73 cm

Fluency

Ex 1

4–5, 6a-c, 7, 8ab, 9, 10, 11ab, 14ab, 15, 16a

4ace, 5, 6cd, 7c-e, 8–10, 11bc, 12, 13, 14bc, 15b, 16ab

4bde, 5b, 6bd, 7c-e, 8bc, 10, 11c, 12, 14cd, 16c

Find the perimeter of each rectangle: a

21 m 6m

21 m

3 cm

10 cm

31 mm

6 mm

Find the perimeter of a rectangle with these dimensions. Draw a diagram if needed: a

31 m

13 m

3 cm

Length: 22 m, Width: 8 m

Length: 2 m, Width: 13 m

Find the perimeter of each square: a

7 cm

18 m

7 cm

5 cm

4 cm

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Find the perimeter of each triangle: a

9 mm

15 cm 8 cm

9 mm

16 cm

14 cm

21 cm

15 cm

9 mm

10 mm

10 cm

8 cm

Find the perimeter of each triangle. Draw a diagram if needed: a

A scalene triangle with side lengths 13 mm, 17 mm and 18 mm.

An equilateral triangle with a side length of 4 m.

An isosceles triangle where the 2 equal side lengths are 13 mm each and the third side measures 4 mm.

Ex 2

A square has a perimeter of 12 mm. Find the side length of the square.

Ex 3

Find the perimeter of this shape:

8 mm

Find the perimeter of these polygons: a

A regular quadrilateral where the length of each side is 7 cm.

A regular pentagon where the length of each side is 6 mm.

A regular octagon where the length of each side is 6 m.

Given the perimeter, P, of these rectangles, ﬁnd the value of the variable: a

b P = 24 m

c P = 20 cm 8 cm

n P = 66 cm

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A rectangle has a perimeter of 26 cm. If it has a length of 9 cm, ﬁnd its width.

Given the perimeter, P, of these triangles, ﬁnd the value of the variable: a

P = 61 cm

P = 29 m 10 m

21 cm

17 cm

d 5 cm P = 16 cm

P = 35 cm 7 cm

Ex 4

Find the side length of these shapes, where P is the perimeter: a

P = 32 cm

P = 20 m

Find the side length of these polygons: a

A regular hexagon with a perimeter of 66 cm.

An equilateral triangle with a perimeter of 21 m.

A regular decagon with a perimeter of 150 mm.

Reasoning

17, 20–22

17–18, 20, 22

18–19, 22–23

Consider a rectangle with a length = 8 cm and a width = 5 cm. Use two different methods to ﬁnd the perimeter of the rectangle.

The perimeter of a square is 24 cm. Will the perimeter increase by 4 times if the length of each side is increased by 4 cm? Explain your reasoning.

The perimeter of a square is 40 cm. Will the perimeter increase by 4 times if the length of each side is increased by 4 times? Explain your reasoning.

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If each side of a rectangle is doubled in length, will this mean the perimeter is doubled? Support this with an example.

A square has a side length of a cm. Write an algebraic expression for the perimeter of the square.

A regular hexagon has a side length of p mm. Write an algebraic expression for the perimeter of the hexagon.

A rectangle has a length of 3x mm and a width of 2x mm. Write an algebraic expression for the perimeter of the rectangle.

Problem-solving 24, 26, 28, 29, 32

25–27, 29, 30, 33, 35

29–34, 36

Find the other side length of a triangle that has a perimeter of 18 m and two side lengths that both measure to be 5 m.

Find the side length of one of the equal sides of an isosceles triangle that has a perimeter of 22 cm and a third side length that measures to be 4 cm.

A rectangular race track has a length of 25 m and a width of 10 m. What distance is covered in one lap of the track?

A 44 cm long piece of wire is bent into the shape of a square. What is the side length of the square?

Hebah is tiling her bathroom with equilateral triangle tiles. She knows the perimeter of each tile, but she wants to ﬁnd the length of each side. If each tile has a perimeter of 36 cm, how long is each side?

A square has the same perimeter as an equilateral triangle. If the triangle has sides of length 8 mm, ﬁnd the side length of the square.

The length of a rectangle is three times its width. Find the dimensions of the rectangle if the total perimeter is 40 m.

Nadia wants to build a thin wire frame for a photo that is 13 cm long and 6 cm wide. How much wire will she need to go around the entire photo?

A company is building a rectangular soccer ﬁeld with a length of 100 m and a width of 60 m. If the cost of fencing per metre is $15, how much will it cost to fence the ﬁeld?

A construction company wants to build a square parking lot with a side length of 120 m. If the cost of fencing per metre is $21, what is the cost of fencing the parking lot?

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A present is contained in a cube shaped box. Ribbon is wrapped around the present as shown in the ﬁrst ﬁgure: a

If the side length of the box is 13 cm, ﬁnd the shortest length of the ribbon needed to neatly go around the box without overlap.

The bow requires 17 cm of ribbon. How much ribbon is needed altogether to wrap the box in this way?

Find the total length of ribbon needed if we wrap the present with two lengths of ribbons, as shown in the second ﬁgure. Assume a single bow is tied.

In a small town, the local park has a rectangular playground area. The length of the playground is represented by the algebraic expression y + 3, while the width is represented by the expression 2x.

y+3

To install a new fence around the playground, the park’s management needs to calculate the perimeter of the area.

Find an algebraic expression for the perimeter of the playground.

Simplify the expression.

A landscape architect is designing a garden that consists of a rectangular ﬂower bed and an adjoining triangular lawn area. The length of the rectangular ﬂower bed is represented by the algebraic expression 4y − 1 3x − 2 and the width is represented by the expression 4y − 1. The triangular lawn has two equal sides with lengths represented by the expression 3y − 2 each.

3x − 2 3y − 2

3y − 2 3x − 2

The landscape architect wants to install a border around the entire garden. a

Find an algebraic expression for the perimeter of the combined garden area.

Simplify the expression.

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12.03 Perimeter of composite shapes After this lesson, you will be able to… • understand the concept of perimeter and how it applies to composite shapes. • determine missing side lengths of composite shapes. • find the perimeter of composite shapes.

Perimeter of composite shapes A composite shape is a figure formed by combining two or more basic shapes, such as rectangles, triangles, circles, etc. To ﬁnd the perimeter of a composite shape, we add the sides around the outside of the shape. Sometimes we need to determine missing side lengths before ﬁnding the perimeter.

Example 1 Find the perimeter of this shape.

12 cm

16 cm

4 cm 8 cm

8 cm

Create a strategy

Apply the idea

Add the lengths of all the given sides.

Perimeter = 8 + 8 + 16 + 12 + 4 = 48 cm

Add the sides Evaluate

Example 2 7m

Consider the figure:

y 2m

17 m

a Find the length x.

Create a strategy The side length x will be equal to the difference between the 5m other two vertical side lengths. Use the fact that the vertical side on the left has the same length as the two vertical sides on the right added together.

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7m x

y 2m

17 m

Apply the idea 5=x+2 5−2=x+2−2 x=3m

Write the relationship in an equation Subtract 2 from both sides Evaluate

b Find the length y.

Create a strategy The side length y will be equal to the difference between the other two horizontal side lengths. Use the fact that the horizontal side on the bottom has the same length as the two horizontal sides on the top added together.

7m 5m

y 2m

17 m

Apply the idea 17 = y + 7 Write the relationship in an equation 17 − 7 = y + 7 − 7 y = 10 m

Subtract 7 from both sides Evaluate

c Calculate the perimeter of the ﬁgure.

Create a strategy The perimeter of a ﬁgure is the sum of all of its sides. Use the results found in parts (a) to (b).

Apply the idea Perimeter = 7 + x + y + 2 + 17 + 5

Add the sides

= 7 + 3 + 10 + 2 + 17 + 5

Substitute x and y

= 44 m

Evaluate

Reflect and check The perimeter can also be worked out by 2 × 17 cm (for the width) +2 × 5 cm (for the height). Can you explain why?

Idea summary To ﬁnd the perimeter of a composite shape, we add the side lengths around the outside of the shape.

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12.03 Practice questions Understanding

Explorer 1–5

Adventurer 2–3, 5

Trailblazer 2–3, 5

Explain how to ﬁnd the perimeter of a composite shape?

Which of these real life scenarios involves the use of perimeter?

Measuring the correct length of fence needed to enclose a garden.

Estimating the amount of ribbon needed to wrap a gift box with known dimensions.

Comparing the length of two different routes to a destination to see which one is shorter.

A construction worker determining the height of a building.

How many sides form the perimeter?

True or false: To find the perimeter of the composite shape we first find the perimeter of the triangle and the perimeter of the rectangle and then we add the two perimeters together.

4 cm

9 cm 12 cm

Starting at the top of the shape and working clockwise, complete the steps to calculate the perimeter of this hexagon: P = 10 + 7 + ⬚ + ⬚ + ⬚ + ⬚ P = ⬚ cm

10 cm 8 cm

7 cm

9 cm

9 cm 11 cm

Starting at the top of the shape and working anticlockwise, complete the steps to calculate the perimeter:

3 cm

P=4+6+3+⬚+⬚+⬚+⬚+⬚ P = ⬚ cm

7 cm

4 cm

6 cm

B 14 cm

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Fluency Ex 1

6–9, 10a, 11ace

6, 8, 10, 11acd

8, 10–11

Find the perimeter of each composite shape: a

9 cm

24 m

8 cm

18 m

18 m 5 cm

8 cm 12 cm

16 cm

6 cm 30 m

4 cm 20 cm

30 m

10 cm

Calculate the perimeter of this ﬁgure:

3 mm x 7 mm 14 mm

Ex 2

Calculate the perimeter of these ﬁgures: a

18 mm 4 mm

13 m

12 mm

4m 16 m

3 mm

y 3 mm

Calculate the perimeter of this ﬁgure:

16 m x

3m y

12 m 14 m

z 3m

Find the perimeter of each figure: a

6 cm

17 cm

12 mm 2 mm

4 mm

3 cm 4 cm

6 cm

5 mm

3 mm

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Find the perimeter of each composite shape: a

2 cm

13 m 4m

8 cm 14 cm 12 m 5 cm 5m

17 m

13 m

6 cm

11 m 9m

35 m

7 cm

10 cm 5 cm 7 cm

14 cm

8 cm

1 cm

6 cm

Reasoning 12

12–13

12–14

Calculate the perimeter of the compound shape by adding the side lengths together.

Find 2 × (17 + 13)

Are your answers to the ﬁrst two parts the same? Explain why or why not.

12–14 9 cm 5 cm 13 cm

17 cm

Use two different methods to calculate the perimeter of this figure:

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Show that the perimeter of the compound shape is the same as the perimeter of a rectangle with sides 6 mm and 5 mm.

2 mm 1 mm 6 mm

Problem-solving 15–17 15

16–19

2 mm

16–17, 19–20

alculate the outside perimeter (shown in blue) of the plot of land on this site plan. C All measurements are given in metres. eet

Cedar Str 127 11.5

Main Stree

Line of wall

Building

17.5 17.5

Property Line 127

b 16

If fencing costs $35.50 per metre, calculate the cost of fencing the plot of land.

A swimming pool has a 2 m wide path around its edge. The outer width of the path is 20 m and the length is 30 m. a

Find the dimensions of the pool.

Calculate the perimeter of the pool.

20 m

30 m

Alexander draws the rectangular plot of land which contains his house and garden. a

Find the perimeter of the plot of land.

Find the perimeter of the house.

Find the perimeter of the garden.

18 m 11 m

Garden 23 m House

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A rectangular swimming pool measures 10 m by 5 m. If a 3 m wide walkway is constructed around the outside of the entire pool, calculate the perimeter of the pool and walkway combined.

A farmer wants to fence a rectangular ﬁeld that measures 50 m by 30 m. The fence is to be placed 2 metres inside from the edge of the ﬁeld.

Calculate the total length of the fence required.

If fencing costs $18.75 per metre, calculate the cost of fencing the ﬁeld.

A square tablecloth measures 2 m by 2 m. A decorative trim of width 10 cm is added around the edges of the tablecloth. a

Calculate the perimeter of the tablecloth and trim combined.

If trim costs $5.40 per metre, calculate the cost of trimming the tablecloth.

12.04 Units of area After this lesson, you will be able to… • describe and provide examples of different units of area measurement and correlate these units with real-life objects or spaces. • select the most appropriate unit of area for measurement. • convert a unit to a different unit of area as required.

Units of area When we have a 2D shape, the amount of space inside the shape is called the area of the shape. The area is a measure of how many squares ﬁt inside the shape. In the metric system the squares to measure area have side lengths of 1 mm, 1 cm, 1 m or 1 km. From smallest to largest, the common units used for measuring area are: • square millimetres (mm2) • square centimetres (cm2) • square metres (m2) • square kilometres (km2)

The unit square

1 unit

1 unit2 of area 1 unit

To be able to visualise and estimate areas, it can be useful to identify common objects that have an approximate area of these units. For example: • 1 mm2 is about the size of a piece of glitter. • 1 cm2 is about the area of a ﬁngernail. • 1 m2 is roughly the area of the floor below a shower or that of a school desk. • 1 km2 is the surface area of 800 Olympic swimming pools or the area of several city blocks.

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A hectare is a special unit of area often used to measure the area of a piece of land. It is equal to a piece of land that is 100 m long by 100 m wide. Therefore 1 ha = 10 000 m2. It’s also still common to see the word acre used to measure the size of land in Australia. 1 hectare is approximately equivalent to 2.47 acres.

Example 1 Choose the most appropriate unit of measurement. a Which of these areas is best measured in square metres (m2): A Your classroom

An exercise book

A matchbox

Australia

Create a strategy Think about what units would be appropriate to measure each object.

Apply the idea Square metres (m2) is suited for big sizes, so we could use it to measure the area of your classroom. Square centimetres (cm2) is suited for smaller objects, so we could use it to measure the area of a matchbox or an exercise book. Square millimetres (mm2) is suited for tiny objects, so we could use it to measure the area of a matchbox. Square kilometres (km2) is suited for bigger places, so we could use it to measure the area of Australia. The answer is option A.

b Which of these areas is best measured in square centimetres (cm2): A Your street

A school playground

Your pencil case

Japan

Create a strategy Think about what units would be appropriate to measure each object.

Apply the idea Square metres (m2 ) is suited for big sizes, so we could use it to measure the area of your street and a school playground. Square centimetres (cm2) is suited for smaller objects, so we could use it to measure the area of your pencil case. Square kilometres (km2 ) is suited for bigger places, so we could use it to measure the area of Japan. The answer is option C.

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Idea summary The area of a 2D shape is the amount of space inside the boundaries of the shape. Area is measured in square units: mm2, cm2, m2 and km2. A hectare is a special unit of area often used to measure the area of a piece of land. 1 ha = 10 000 m2.

Estimate area Estimation of areas and understanding area benchmarks are useful skills that will help us to be able to appropriately choose units for area and to be able to visualise the size of something if we are told the units of area. There are some a few different ways we can estimate area: • Use a mental image of common objects of size 1 mm2, 1 cm2, 1 m2 and/or 1 km2 and try to picture how many it would take to cover the object. • For shapes that are rectangular (or close to rectangular), estimate the length and width of the object and then multiply: Area = length × width. • If a picture of the item is available, you could draw a scale grid over it and estimate by counting the number of squares needed to cover the item.

Example 2 Estimate the total area of the artwork above the sofa if the sofa is 2.2 m in length. A 1.7 m2

0.02 m2

C 5 m2

0.4 m2

2.2 m

Create a strategy Use the dimensions of the sofa to estimate the area of the artwork above it.

Apply the idea Let’s start by estimating the length of the artwork. Since the length of the sofa is 2.2 m, 2 m is a reasonable assumption for the length of the artwork. We could estimate the width of the artwork to be half of the length. So the largest area the artwork could take up would be 2 m × 1 m = 2 m2. This rules out option C. The closest to 2 m2 is option A at 1.7 m2.

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Example 3 Each square on the grid has an area of 3 mm2. Estimate the area of the curved shape shown: A 111 mm2

72 mm2

C 26 mm2

48 mm2

Create a strategy

Apply the idea

Count the number of squares that are inside or almost completely inside the curved shape, multiply by the area of 3 mm2 and then choose the option that is closest to this result.

Area = 21 × 3 = 63 mm2

Multiply 21 by 3 Evaluate

Since option B is the nearest and higher than 63 mm2, the estimated area of the curved shape is 72 mm2.

Idea summary Techniques we can use to estimate area: • Use a mental image of common objects of size 1 mm2, 1 cm2, 1 m2 and/or 1 km2 and try to picture how many it would take to cover the object. • For shapes that are rectangular (or close to rectangular), estimate the length and width of the object and then multiply: Area = length × width. • If a picture of the item is available, you could draw a scale grid over it and estimate by counting the number of squares needed to cover the item.

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Convert units of area When we convert units of area we are changing the size of the squares that the area is measured in.

Interactive exploration

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The conversion factors are summarised in the conversion chart: ÷ 100

mm²

× 100

÷ 10 000

cm²

÷ 10 000

hectares

m²

× 10 000

hectares

acres

× 2.47

Example 4 Convert 6 m2 to cm2.

Create a strategy Sketch a square with side length of 1 m. We want to convert to centimetres, how many centimetres will each side length be and work out how many square centimetres is in 1 square metre.

Apply the idea 6 m2 = 6 × (100 × 100) 2

= 60 000 cm

Convert the square metre into centimetres Evaluate

Idea summary When converting between units of area: • Multiply if converting to a smaller unit - more smaller squares will be needed to cover the same area. • Divide if converting to a larger unit - fewer larger squares will be needed to cover the same area.

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12.04 Practice questions Understanding

Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

Fill in the blanks.

There are ⬚ mm in 1 cm, ⬚ cm in 1 m and ⬚ m in 1 km.

Order these units of area from smallest to largest: • km2 • cm2 • mm2 • m2

Name an object that has an area of approximately 1 m2.

Name an object that has an area of approximately 500 cm2.

Fluency Ex 1

4, 6–9, 11ab, 12

4, 6–9, 11bc, 12

4–6, 8, 10, 11cd, 12

For these areas, choose the most appropriate unit of measure from the list: • Hectares • Square centimetres • Square metres • Square kilometres a

Postage stamp

Bathroom wall

Paper

Cabinet

City

Television

Kitchen

Farm

Netball court

Table

National park

Door

Find the area of a square with sides 290 cm in length. Give your answer in square metres.

This garden bed has three rows of four plants. Each plant is spaced about 24 cm apart and sits 12 cm away from the edge of the garden bed. Estimate the area of the garden bed in square centimetres.

Ex 2

Estimate the area of the Mathspace logo on the box. A

150 cm2

1 m2

1000 cm2

750 cm

50 cm 1.5 m

Ex 3

50 cm

Each square on the grid has an area of 4 mm2. Estimate the area of the curved shape.

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16 14 12 10 8 6 4 2 0

Estimate the area of the shape shown if each axis is measured in metres.

Estimate the area of the butterﬂy if each axis is measured in millimetres.

Estimate the area: a

10 cm2

50 cm2

1 m2

b 10 cm 4 cm

10.4 9.1 7.8 6.5 5.2 3.9 2.6 1.3 0

200 cm2

55 000 mm2

6 m2

6000 cm2

7000 mm2

2 m2

100 mm

7000 cm2

6000 mm

7 m2

70 cm2

400 mm2

2000 mm2

78 cm

2 4 6 8 10 12 14 16

1 2 3 4 5 6 7 8

70 cm

18 mm

66 mm Ex 4

Convert: a

3 m2 to square centimetres 2

7 m2 to square centimetres

0.56 km to square metres

6 ha to square metres

10.4 ha to square metres

20 000 m2 to hectares

Reasoning

13, 15, 16

13, 14, 16

There are 100 cm in 1 m but to convert from square metres to square centimetres we must multiply by 10 000. Explain why this is correct.

Determine whether these statements are true or false. Explain your reasoning. a The unit of estimation must always be the same as the unit of measurement used to ﬁnd the exact area. b The area of a square is equal to the perimeter of the square divided by 4. c The area of a rectangle with length 5 cm and width 2 cm is equal to the area of a rectangle with length 2 cm and width 5 cm. d The area of a shape can never be greater than its perimeter. e When estimating area, it is always best to round the measurements to the nearest whole number.

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Why is it a good idea to estimate the area of a shape before calculating it exactly?

A hectare is approximately the size of an athletics arena. Why are blocks of land, farms and large properties measured in hectares?

Problem-solving 17–19

17–22

17–24

These road signs are pictured with relative sizings. If the GIVE WAY sign is an equilateral triangle with a side length of 75 cm, estimate the area of the BICYCLES EXCEPTED sign.

A park has an area measuring 19 acres. Given that 1 acre is approximately 4000 m2, ﬁnd the area of the park in square metres.

A sample of rectangular farms around Australia were measured and their dimensions, in metres, are recorded in the table:

Find the area of each farm in square metres.

Which farms have an area of more than 1 ha?

Which farms have an area of less than 1 ha?

Which farm has an area of exactly 1 ha?

Farm Length, m Width, m Area, m2 1 200 100 2 350 21 3 100 24 4 400 40 5 100 100

A property covers an area of 7.22 ha. Given that 1 ha is 10 000 m2, calculate: a

The area of the property in square metres.

The area of the property in acres, given that 1 acre is approximately 4000 m2.

A large cattle station covers an area of 320 acres. Given that 1 acre is approximately 0.4 ha and 1 ha is 10 000 m2, calculate: a

The area of the cattle station in hectares.

The area of the cattle station in square kilometres.

A farm covers an area of 51 000 m2. Given that 1 ha is 10 000 m2 and 1 acre is approximately 4000 m2, calculate: a

The area of the farm in hectares.

The area of the farm in acres. Round your answer to two decimal places.

Bob owns a paddock with an area of 290 acres. He wants to advertise his eggs as free range, with no more than 400 chickens per hectare. Given that 1 acre is approximately 0.4 ha, ﬁnd the maximum number of chickens that Bob can keep in his paddock.

A farm covers an area measuring 340 acres. Given that 1 acre is approximately 0.4 ha and 1 ha is 10 000 m2, calculate: a

The area of the farm in hectares.

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12.05 Area of rectangles and squares After this lesson, you will be able to… • understand the concept of area and square units. • find the area of rectangles and squares using the formula A = l × w or A = s2. • use the formula to find a missing length or width given the area of a rectangle or square. • solve problems involving area of rectangles and squares.

Area of rectangles and squares The area of a rectangle is the amount of space inside the rectangle. We measure area in square units: 1 square millimetre = 1 mm2 1 square centimetre = 1 cm2 1 square metre = 1 m2 1 square kilometre = 1 km2

2 cm 4 cm

This rectangle contains 8 squares, each with a side length of 1 cm. So this rectangle has an area of 8 cm2. Another way to ﬁnd the area is to multiply the length and width, 4 × 2 = 8 cm2.

The area of a rectangle is given by: Area = length × width A=l×w

The area of a square is given by: Area = side × side A = s × s = s2

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Example 1 Find the area of this rectangle: 9 cm 12 cm

Create a strategy

Apply the idea

Use the formula for the area of a rectangle.

A=l×w

Write the formula

= 12 × 9 = 108 cm

Substitute the values 2

Evaluate

Example 2 Find the area of this square: 8 cm

Create a strategy Use the formula for the area of a square.

Apply the idea A=s×s

Write the formula

=8×8 = 64 cm

Substitute the values 2

Evaluate

Idea summary To ﬁnd the area of a rectangle: A=l×w A l w

is the area of the rectangle is the length of the rectangle is the width of the rectangle

To ﬁnd the area of a square: A=s×s A is the area of the square s is the side length of the square

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Find missing length or width given the area A rectangle has an area of 28 km2 and a length of 7 km. How can we determine the width of the rectangle? Area = length × width, so 28 = 7 × width. Therefore, the width is given by 28 ÷ 7 = 4, so the width of the rectangle is 4 km. 28 km²

7 km

? km

Example 3 Find the width of this rectangle if its area is 66 m2 and its length is 11 m.

Area = 66 m²

11 m

Create a strategy

Apply the idea

Use the formula for the area of a rectangle.

Write the formula

Substitute the values

Divide both sides by 11

Evaluate

Idea summary To ﬁnd a missing length or width of a rectangle, substitute the area and given dimension into the formula: A = l × w and then solve for the unknown variable.

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12.05 Practice questions Understanding 1

Adventurer 2, 4–5, 7bc

Trailblazer 5, 7bd

In these diagrams, each grid square represents 1 square unit. Find the area of each shape by counting the number of grid squares it covers: a

Explorer 1–3, 4a-d, 5–6, 7ac

The area of this shape was measured using a centimetre grid. What unit should we use to describe its area?

For each pair of shapes: i

Find the area of Shape A.

iii

Which shape is smaller?

Shape A

Shape B

Find the area of Shape B.

Shape A

Shape B

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For each pair of shapes: i

Find the area of Shape A.

iii

Which shape is larger?

Shape A

Shape B

Shape A

Shape B

Shape A

Shape B

Shape A

Find the area of Shape B.

Shape B

Shape A

Shape B

Order the three shapes’ areas from smallest to largest.

Shape A

Which shape is larger?

Shape A

Shape B

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Shape B

Shape C

What is the length and width of the rectangle?

Find the area of the rectangle.

8–9, 10ac, 11ab, 12, 14, 15a

Fluency Ex 1

8b-e, 9bc, 10a, 11–13, 15

8de, 9bc, 10bd, 12–14, 15b

Find the area of these rectangles: a

4 cm 10 m 6 cm

e 23 mm 6 mm

10 m

9 mm

10 m

6 mm 23 mm

3 mm

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Ex 2

Find the area of each square: a

9 mm

18 m

9 mm

4 cm

9 mm

For each rectangle: i

Find the perimeter.

3 cm

Find the area.

4 cm 8 cm

2 cm

6 cm

d 3 cm

5 cm

7 cm

Find the area. Draw a diagram if needed. a

A square whose side is 7 cm.

A rectangle whose length is 10 cm and width is 3 cm.

A rectangle whose length is 12 cm and width is 5 cm.

Find the side length of this square:

A = 25 mm²

A square has a perimeter of 32 cm. a

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Find the side length of the square.

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Find the area of the square.

Ex 3

Find the length of the missing side of this rectangle:

Area = 30 mm²

5 mm

Find the length of the missing side in these rectangles: a

A rectangle that has an area of 36 m2 and a width of 4 m.

A rectangle that has an area of 18 m2 and a length of 9 m.

Reasoning 16

16–17, 19

16–19

16–21

Find the area of the square.

Double the side lengths of the square and ﬁnd the area.

When we double the side lengths of a square, do we also double the area of the square?

7 cm

Investigate whether doubling each side of the rectangle also doubles the area of the rectangle.

From the results of the last two questions: a

When the sides of the square or rectangle were doubled, by what factor was the area increased?

Explain why the area is increased by this number.

If the sides of a square or rectangle were multiplied by 5, by what factor would the area increase?

If the sides of a square or rectangle were multiplied by x, by what factor would the area increase?

Determine a quadrilateral that has its area equal to its perimeter.

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Explain how the area of a rectangle could be used to ﬁnd the area of this triangle. 55 cm 48 cm 73 cm

For a rectangle with an area of A = 50 m2: a

Complete the table, showing the possible dimensions of rectangles with an area of A = 50 m2. Length ⬚ ⬚

Width ⬚ ⬚

Area 50 m2 50 m2

Length ⬚ ⬚

Width ⬚ ⬚

Area 50 m2 50 m2

Complete the table, showing the perimeter of each rectangle. Perimeter ⬚ ⬚

Using the table from part (b), explain why some rectangles with the same area cannot have the same perimeter.

Problem-solving 22, 24–25

23–24, 26

23, 25–28

A rectangular swimming pool has a length of 6 metres and a width of 4 metres. What is the area of the pool in square metres?

A square piece of cardboard is cut into 4 smaller squares, each with an area of 4 square centimetres. What is the area of the original square cardboard in square centimetres?

A company has a rectangular warehouse that measures 50 metres by 30 metres. They want to install a new ﬂoor in the warehouse that costs $15 per square metre. What will be the total cost of the new ﬂoor?

A wall in a room has a rectangular shape with a length of 3 m and a height of 2.5 m. If each square metre of wall needs one litre of paint and the paint costs $20 per litre, what will be the cost of painting the wall?

A bathroom tile has a square shape with an area of 0.16 square metres. If a bathroom ﬂoor measures 6 metres by 4 metres, how many tiles are needed to cover the entire ﬂoor?

A square painting is mounted in a frame that is 3 centimetres wide on all sides. If the painting has an area of 36 square centimetres, what is the total area of the painting and frame?

Emma wants to lay new tiles in her kitchen which measures 4.5 m by 3.8 m. The tiles she plans to use come in boxes of 20 tiles, each measuring 30 cm by 30 cm. How many boxes of tiles will Emma need to cover her kitchen ﬂoor?

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12.06

Area of triangles

After this lesson, you will be able to… • understand the relationship between triangles and rectangles in terms of their areas. • calculate the area of a triangle using the formula , where b is the base of the triangle and h is its height. • find the missing base or height of a triangle when given its area.

Triangles and rectangles A rectangle can be cut into two equal triangles. In this diagram, we say the base of the triangle is 8 m and the perpendicular height is 5 m.

Interactive exploration

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Idea summary The area of a triangle is half of the area of a rectangle with the same height and base.

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Area of a triangle The area of a triangle is given by

Example 1 Find the area of this triangle:

7 cm

10 cm

Create a strategy Use the formula for the area of a triangle.

Apply the idea Write the formula

Substitute b = 10 and h = 7

Evaluate

Example 2 Find the area of the given triangle with a base length of 10 metres and a perpendicular height of 8 metres.

10 m

Create a strategy Use the formula for the area of a triangle.

Apply the idea For this triangle, the base is b = 10 m and the height is h = 8 m. Write the formula

Substitute b and h

Evaluate

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Idea summary The area of a triangle is given by:

A is the area of the triangle b is the base of the triangle h is the height of the triangle

Find base or height given area We can use the rule for the area of a triangle to ﬁnd a missing dimension.

? km

This triangle has an area of 45 km2 and a base of 9 km. How can we determine the height of the triangle?

We know that Now, multiply

9 km 45 km²

, which means that

by 9 to give 4.5, which means that 45 = 4.5 × height.

To ﬁnd the height, we divide 45 by 4.5 to give 10. So, the height of the triangle is 10 km.

Example 3 Find the value of h if the area of this triangle is 48 m2. hm

Create a strategy

Apply the idea

Use the area of a triangle formula and solve for h.

Write the formula

Substitute A = 48 and b = 4

Evaluate the multiplication

Divide both sides by 2

Evaluate

Therefore, the height is 24 m.

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Idea summary To find a missing base or height when given the area, we can use the formula for the area of a triangle.

12.06 Practice questions Understanding 1

Explorer 1, 2

Adventurer

Trailblazer

1, 2

What fraction of the rectangle is the shaded triangle?

If the area of the rectangle is 50 cm2, ﬁnd the area of the triangle.

If the area of the shaded triangle is 10 cm2, what is the area of the rectangle?

width

length

Identify the base and height of each triangle: a

3 cm

36 mm

5.3 cm 2.7 cm 20 mm

15 mm

22 mm

6 cm

8 cm

6.4 cm

10 cm

12 cm

19.5 cm

29 cm

14.5 cm 12.5 cm

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3a-c, 4, 5a-c, 6ac, 7, 8ab, 9a, 10

Fluency 3

3b-d, 4, 5a-d, 6ac, 7, 8ac, 9ac, 10

3d-f, 4, 5c-f, 6bd, 7, 8cd, 9bd, 10

For each figure: i

Find the area of the entire rectangle.

Find the area of the triangle.

5 cm

3 cm

10 cm

d 8m

2 cm 11 cm

10 m

8 cm

16 m

3 cm 4m Ex 1, 2

Find the area of each triangle: a

5m 6 cm 12 m 7 cm

55 cm 48 cm

8 cm

73 cm

6 cm

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Find the area of a triangle which has these dimensions: a

Ex 3

Height = 8 cm and base = 9 cm

Height = 4 m and base = 6 m

Height = 7 mm and base = 8 mm

Height = 3 cm and base = 9 cm

Base = 12 cm and height = 100 mm

Base = 5.2 m and height = 200 cm

Determine whether these dimensions could form a triangle with an area of 20 mm2: a

Base = 10 mm, height = 4 mm

Base = 1 mm, height = 20 mm

Base = 5 mm, height = 8 mm

Base = 2 mm, height = 2 cm

For this triangle with an area of 20 mm2, find the value of b.

5 mm

b mm

Find the base measurement for these triangles given the area and height: a

Area = 18 cm2 and height = 9 cm

Area = 40 cm2 and height = 10 cm

Area = 400 mm2 and height = 4 cm

Area = 12.5 mm and height = 10 mm

Find the height measurement for these triangles given the area and base: a

Area = 90 cm2 and base = 20 cm

Area = 150 cm2 and base = 50 cm

Area = 2500 mm2 and base = 50 cm

Area = 33.75 mm and base = 4.5 mm

Complete this table of base and height measurements for four different triangles, which all have an area of 30 m2:

Reasoning 11

11, 14

Base (m) Height (m) Area (m2)

10 12 30

11, 12, 14, 16

Explain how the area of a triangle is connected to the area of a rectangle using this diagram:

2 30

12, 13, 15, 16

A right-angled triangle has a base of 5 cm and a height of 10 cm. a

If the height is halved, what effect does it have on the area of the triangle?

If the base is doubled, what effect does it have on the area of the triangle?

Two right triangles have the same area. If one triangle has a base of 8 cm and a height of 6 cm, what are the possible dimensions of the other triangle, given the base and height are whole numbers?

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If two right-angled triangles have the same area, does this mean they have exactly the same shape? Explain your reasoning.

A triangle has base k units and perpendicular height p units.

Write an algebraic rule to ﬁnd A, the area of the triangle.

If p = 5 mm and k = 20 mm, calculate the area of the triangle.

The formula to ﬁnd the area of a triangle a

Rearrange the formula to make b the subject (make it b = ).

Rearrange the formula to make h the subject.

Problem-solving 17, 18, 20, 21

18, 19, 21

19, 21–24

A farmer wants to install a triangular shade sail in her paddock to protect her sheep from the sun. The base of the shade sail is 12 m long and the height of the triangle is 8 m. What is the area of the triangular shade sail in square metres?

A civil engineer needs to design a triangular traﬃc island for a new road intersection. The base of the triangle is 20 m long and the height of the triangle is 15 m. Calculate the area of the triangular traﬃc island in square metres.

An artist is painting a triangular canvas with a base of 80 cm and a height of 50 cm. Calculate the area of the canvas that needs to be painted: a

in square centimetres

in square metres

An architect is designing a triangular courtyard for a new apartment building. The base of the triangle is 30 m long and the height of the triangle is 20 m. a

Calculate the area of the courtyard.

If paving tiles cost $25 per square metre, calculate the cost of paving the courtyard.

A painter charges $30 per square metre to paint walls. If a room has a triangular section with a base of 8 m and a height of 5 m, how much will it cost to paint that section of the wall?

Joy wants to paint her living room walls. One of the walls measures 3 m high and 5 m wide, while the other wall measures 3 m high and 4 m wide. There is a triangular window on one of the walls that has base of 2 m and height of 1 m. If each litre of paint can cover an area of 6 m2, how many exact litres of paint will Joy need to buy?

Cody is re-tiling his rectangular bathroom ﬂoor. The bathroom measures 2 m by 3 m, but there is a shower cubicle in the corner in the shape of a right triangle with a base of 1 m and a height of 1.5 m. Each tile measures 20 cm by 20 cm. a

Calculate the number of tiles that Cody needs to buy.

If Cody needs to account for 10% wastage, how many tiles does he need to buy?

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A gutter running along the roof of a house has a cross-section in the shape of a triangle: If the area of the cross-section is 40 cm2 and the length of the base of the gutter is 10 cm, ﬁnd the perpendicular height, h, of the gutter.

h cm

10 cm

12.07 Area of composite shapes After this lesson, you will be able to… • identify composite shapes and visualise them as a combination of simple shapes. • add the areas of simple shapes to find the total area of a composite shape. • apply subtraction to find the shaded or remaining area of composite shapes with holes and cutaways. • solve real-world problems involving composite shapes by applying your knowledge of finding area.

Area of composite shapes using addition We can think of a composite shape as one that is made from a number of smaller shapes. Dashed lines can be used to visualise which simple shapes make up a composite shape. We ﬁnd the total area by adding the areas of each of the simple shapes.

Example 1 Find the area of this figure: 4 mm

9 mm

15 mm

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Create a strategy Calculate the area of the triangle and the rectangle, then add them.

Apply the idea Area of the triangle: Write the formula

Substitute b = 15 and h = 4

Evaluate

Area of the rectangle: A=l×w

Write the formula

= 15 × 9

Substitute l = 15 and w = 9

= 135

Evaluate

Total area: A = 30 + 135 = 165 mm

Add the areas Evaluate

Example 2 Find the total area of this figure: 6 cm

15 cm

19 cm

6 cm

Create a strategy Divide the shape into three. Use the area formulas for a parallelogram and rectangle.

6 cm 15 cm

19 cm

6 cm

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Apply the idea Note that both the parallelograms have the same dimensions, so they will have the same area. Area of parallelograms: A = b × h Write the formula = 19 × 6

Substitute b = 19 and h = 6

= 114

Evaluate

Both parallelograms = 114 × 2 = 228

6 cm 19 cm

Multiply the area by 2 Evaluate

Area of rectangle: A=l×w

Write the formula

= 19 × 15

Substitute l = 19 and w = 15

= 285

Evaluate

15 cm

19 cm

Total Area: A = 228 + 285 = 513 cm

Add the areas Evaluate

Idea summary To ﬁnd the area of a composite shape, we can add the areas of the individual shapes it is made up of.

Area of composite shapes using subtraction We can also ﬁnd the area of a composite shape using subtraction.

Area of = composite shape

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−

Area of rectangle

−

Area of triangle

Example 3 Find the shaded area of this figure:

6 cm 3 cm 14 cm

Create a strategy Subtract the area of the triangle from the area of the parallelogram. −

Apply the idea Area of triangle: Write the formula

Substitute b = 14 and h = 3

Evaluate

Area of parallelogram: A=b×h

Write the formula

= 14 × 6

Substitute b = 14 and h = 6

= 84

Evaluate

Area of composite shape: A = 84 − 21 = 63 cm

Subtract the area of triangle from the area of parallelogram Evaluate

Idea summary To ﬁnd the area of a composite shape with holes and cutaways, we can subtract the cutaway areas from the larger area.

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12.07 Practice questions Understanding 1

Explorer

Adventurer

1, 2

2, 3

Trailblazer 2, 3

For this composite shape: 6 cm

5 cm 9 cm

Name the two geometric shapes the composite shape is comprised of.

Explain how to ﬁnd the area of the shape.

Explain how to ﬁnd the area of this composite shape. 6 cm

8 cm

10 cm

Explain how to ﬁnd the area of this composite shape: 2m 7m 13 m

500

Using addition method

Using subtraction method

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Fluency Ex 1

4, 6, 7, 8a-d, 9, 10ab

5, 7–9, 10b-d

5, 7, 8c-g, 10cd

For this composite shape: 16 cm

14 cm

16 cm

Ex 2

8 cm

Find the area of the triangle.

Find the area of the rectangle.

Find the total area of the composite shape.

The quadrilateral in the diagram has a diagonal measuring 14 m.

Calculate the total area of this quadrilateral.

14 m

For this composite shape:

4 cm

Determine the dimensions of rectangle A.

Find the area of rectangle A.

Find the area of rectangle B.

Calculate the total area of the given shape.

7 cm 6 cm

A 3 cm

B 14 cm

For each figure: i

Find the area of shape A.

Determine the dimensions of shape B.

iii

Find the area of shape B.

Calculate the total area of the composite shape.

6 cm

17 cm

12 mm 2 mm

B 4 mm

3 cm 4 cm

6 cm

5 mm

3 mm

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Find the area of each composite shape: a

2 cm

13 m

15 mm 2 mm

8 cm

14 mm 14 cm

12 m 4 mm

5 cm

10 cm

3 cm

8 cm

5 cm 7 cm

7 cm

10 cm

14 cm 2 cm

8 cm 8 cm

6 cm

16 cm

8 cm 11 cm 4 cm 13 cm

Ex 3

For these figures: • Figure 1:

•

Figure 2:

• Figure 3: 13 cm

13 cm 4 cm 9 cm

4 cm

9 cm

9 cm 6 cm 6 cm

502

Find the area of the shaded rectangle in Figure 1.

Find the area of the shaded triangle in Figure 2.

Find the area of the shaded region in Figure 3.

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Find the area of the shaded region in each figure: a

18 m

12 m 12 cm 5m 4 cm

12 m

4m 5m 15 m

9 cm

6 cm 16 cm

Reasoning 11

11, 12

11–13

11, 13

Calculate the area of this composite shape using two different methods: 5m

9m 5m 13 m

Using addition method

Using subtraction method

Calculate the area of this composite shape using the most efficient method. Explain your reasoning. 13 m

17 m 11 m

35 m

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The aim is to calculate the shaded area in Figure 1. Use the images in Figures 2, 3 and 4 to write a set of steps explaining how to do this. Note that you do not need to calculate the actual areas. • Figure 1: • Figure 2: 15

3 7

11 4 5

• Figure 3: •

Figure 4: 15

7 4 5 5

Problem-solving 14, 16, 17, 19 14

15, 17, 18, 20

15–18, 20, 21

The dining room ﬂoor is to be tiled with slate tiles. The cost of tiling the room is $10 per square metre. a

Calculate the area to be tiled.

How much will it cost to tile the room?

9m 6m 8m

A children’s book is in the shape of a house. Calculate how much wrapping paper is required to wrap the book as a present, assuming the width of the book is negligible.

4 cm

9 cm 12 cm

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Scott draws the plot of land which contains his house and garden.

18 m

Find the total area of the plot of land.

Find the area covered by the house.

Find the area covered by the garden.

If the cost of turﬁng the garden is $26 per square metre, how much will it cost Scott to turf the garden?

23 m House

11 m

Garden

A swimming pool has a 2 m wide path around its edge. The outer width of the path is 20 m and the length is 30 m.

20 m

Calculate the area of the surface of the pool.

Calculate the area of the path around the pool.

Calculate the cost of a pool blanket if blankets cost $5.55 per square metre.

30 m

A farmer needs to fertilise a rectangular field that measures 500 m by 300 m. The field contains a square shed with dimensions 25 m by 25 m. Fertiliser costs $0.03 per square metre. Calculate the cost of fertilising the field.

A graphic designer designs a logo for a company. If A4 paper is approximately 21 cm × 30 cm, calculate the amount of white space that will be left after the logo is printed.

3 cm

8 cm

6 cm 11 cm

A triangular backyard is to have two smaller garden beds planted in it. The rest of the backyard will be covered with crushed rock.

20 cm

17 m

Find the area of the crushed rock.

If crushed rock costs $12 per square metre, calculate how much it will cost to cover the backyard.

A farmer is going to fertilise his paddock which has this shape: a

Find the area of the paddock.

If it takes 150 kg of fertiliser to fertilise 100 square metres, how many kilograms of fertiliser should the farmer purchase?

5m 10 m 20 m 52 m 47 m

201 m

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MODELLING INVESTIGATION

Solar panels Explore how mathematics can help you plan a solar installation for a community shed. Initially you will be asked to decide on the shed roof size, the solar panel size, then determine how many panels will fit onto the roof and finally calculate how much energy you will generate in a day. An average day in Western Australia will have 7 hours of energy producing light.

Objectives • To analyse the solar panel situation and decide what is needed to plan it. • To use mathematics to show how many panels fit and how much energy you will produce. • To explain your plan and how your choices affect the outcome.

Analyse the solar panel situation You are planning a solar installation for a shed. Pick your own shed roof size, solar panel size and determine how much energy you will generate each day. Research typical solar panel sizes and their daily energy output.

Interactive exploration

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Investigate 1.

State the dimensions of your shed and solar panels. Will you use exact numbers or approximations? Why?

What might affect your plan? List one assumption and one thing that limits you.

How many panels fit on your sketched roof? Does it match what you would expect from your sizes?

Were there any challenges in sketching the layout? How might you improve your method?

What difference in energy production would you expect if your shed was located in Albany compared to Karratha?

Steps to plan your solar panel layout: Draw a rectangle on paper to represent your shed roof (for example 6 m × 4 m) using a ruler. Draw a smaller rectangle for your solar panel (for example 1 m × 0.5 m). Sketch how many panels fit on the roof by repeating the panel shape inside the roof outline.

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Your task is to create a physical sketch of your solar panel layout based on your chosen sizes. Use the steps above as a guide. For example, start with a shed roof of 6 m length and 4 m width and panels of 1 m × 0.5 m. 1m

0.5 m 6m

0.5 m

Represent with mathematics Use mathematics to show how many solar panels fit on the shed roof and how much energy you will generate in a day.

Interactive exploration

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Investigate 1.

Show how you will calculate the number of panels that fit on the roof using your sizes.

Show how you will calculate the daily energy output using your sunlight hours and energy rate.

Write your method as a number sentence or list of steps.

Interpret your plan Look at your results and explain what they mean for the solar panel installation.

Investigate 1.

Do your number of panels and energy output seem realistic for the shed? Why or why not?

Change one variable or assumption. How does it change your total?

Explain your solar panel plan to the class. How do your assumptions affect it?

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Discussion 1.

What was the hardest part of planning the solar panel installation with mathematics? Why?

How could you make your plan better if you had more time or tools?

Talk to a friend about their plan. How was it different from yours?

Did you know?

Solar panels work best when angled correctly! In Australia, they usually face true north to capture the most sun. Using geometry and trigonometry, designers find the ideal tilt — too flat or too steep reduces efficiency. The right angle means more energy and bigger savings! Engineers also consider shadows from nearby trees or buildings when calculating placement. It’s a smart mix of maths and design that helps power homes more sustainably!

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12 Chapter review 1

Find the area of each shape: a

7 mm

8 cm 7 mm

11 cm

7 mm

Find the length of the missing side in these rectangles: a

A rectangle that has an area of 24 m2 and a width of 3 m.

A rectangle that has an area of 44 m2 and a length of 4 m.

Find the area of these triangles: a

12 cm 5 cm

7 cm 8 cm

Find the base measurement for these triangles given the area and height: a c

Area = 24 cm2 and height = 6 cm 2

Area = 15 mm and height = 5 mm

Area = 35 cm2 and height = 7 cm

Area = 360 mm2 and height = 6 cm

Find the area of these composite shapes: a

3 cm

7 cm

9 cm 15 cm

6 cm

9 cm

15 cm

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Find the area of the shaded region in these figures: a

11 cm

b 13 m

3 cm

6m 16 m

Each square on the grid has an area of 3 mm2. Estimate the area of the curved shape:

An amusement park has an area measuring 22 acres. Given that 1 acre is approximately 3000 m2, find the area of the park in square metres.

A farm covers an area measuring 280 acres. Given that 1 acre is approximately 0.3 ha and 1 ha is 10 000 m2, calculate: a

The area of the farm in hectares

The area of the farm in square metres

A rectangular piece of fabric is divided into 6 smaller rectangles, each with an area of 9 square centimetres. What is the area of the original rectangular fabric in square centimetres, assuming no fabric is wasted?

A landscape architect needs to design a triangular garden for a new public park. The base of the triangle is 30 m long and the height of the triangle is 12 m. Calculate the area of the triangular garden in square metres.

A wallpaper installer charges $25 per square metre to apply wallpaper on walls. If a room has a triangular section with a base of 10 m and a height of 6 m, how much will it cost to apply wallpaper on that section of the wall?

A homeowner plans to install carpet in their bedroom floor. The cost of the carpet installation is $11 per square metre. How much will it cost to install the carpet in the bedroom?

11 m 6m 10 m

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Ryujin needs to apply fertiliser to his rectangular lawn that measures 250 m by 150 m. The lawn contains a square gazebo with dimensions 20 m by 20 m. Fertiliser costs $0.02 per square metre. Calculate the cost of applying fertiliser to the lawn.

A triangular garden is to have a pond and a fountain installed in it. The rest of the garden will be covered with artificial grass. a

Find the area of the artificial grass.

If artificial grass costs $12 per square metre, calculate how much it will cost to cover the garden.

Pond

Fountain 19 m

10 m

4m 8m 22 m

Did you know?

Tilers use area calculations every day to figure out how many tiles are needed for floors and walls. By measuring the length and width of a surface, they can make sure there’s enough material — without too much leftover waste. It also helps them decide how to cut tiles to fit around edges and corners! Area helps bring precision to hands-on trades like construction and renovation!

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Big ideas • Solids can be visualised in a number of ways for different purposes, nets allow for the surface of the solid to be explored and isometric and perspective views provide the viewer with detail about the image. • The volume of a three-dimensional object is the amount of space that it occupies, measured in cubic units.

Curriculum Outcomes • Move flexibly between building and drawing rectangular and composite rectangular prisms from different views • Establish and apply relationships between the number of identical layers of cubic units, the number of cubic units in each identical layer and volume for rectangular prisms and composite rectangular prisms. Generalise and apply formula using appropriate units • Analyse, represent and interpret problems in real-world situations involving perimeter and area of squares, rectangles, triangles and rectangular composite shapes, parallel lines, properties of triangles, transformations of points, views of rectangular prisms and rectangular composite objects, volume and/or Australian time zones

13 Solids and volume Chapter outline 13.01 Nets of solids 13.02 Visualise solids 13.03 Volume using unit cubes 13.04 Units of volume 13.05 Volume of right prisms 13.06 Attributes in measurement formulas Chapter 13 review

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The Great Pyramid of Giza’s volume is about 2.5 million cubic metres!

13.01

Nets of solids

After this lesson, you will be able to… • identify nets for different solids, including cubes, rectangular prisms, triangular prisms and pyramids.

Nets We can never see every part of a three-dimensional object at once - there is always part of it that is behind the view we are looking at.

Interactive exploration

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To better think about a solid object, we sometimes represent it with its net. Each face of the solid is laid flat on the same surface, breaking it along the edges and folding it out. This way, we can think about folding it back up along its edges to recover the original shape. There are many ways to unfold a net from a solid. We will investigate nets of prisms and pyramids. Here are some prisms: Prisms

Triangular

Square

Rectangular

Pentagonal

Hexagonal

Octagonal

Prisms have rectangular sides and the shape on the top and the base is the same. The name of this shape gives the prism its name. Here are some pyramids: Pyramids

Triangular

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Square

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Pentagonal

Hexagonal

Octagonal

Pyramids have triangular sides and the shape on the base gives the pyramid its name.

Example 1 Choose the net that folds to give this shape:

Create a strategy

Apply the idea

Choose the net that has the same number of faces as the solid and can be folded up along its edges to make the solid.

The answer is option D.

Example 2 Choose the shape that has this net:

Apply the idea Since there is only one square, it must be a square pyramid. The answer is option B.

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Idea summary The net of a prism or a pyramid is created by unfolding the solid and laying each face flat. Prisms have rectangular sides and the shape on the top and the base is the same. Pyramids have triangular sides. The shape of the base gives the prism or pyramid its name.

Cross-sections of prisms and pyramids Interactive exploration

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When we cut a prism with a flat slice parallel to its base, we’ll get a shape that’s the same as the base and the same size. This is the prism’s cross-section. When we are finding the cross-section of a pyramid, we’ll also get a shape that’s the same as the base, but it will be smaller.

Example 3 Consider this solid:

If the solid is cut straight down below the dotted line, what will be the shape of the cross-section? A A pentagon

A triangle

A hexagon

A square

Create a strategy

Apply the idea

If the solid is cut straight down below the dotted line, the shape of the cross-section will be the same as the front face.

The front face of the solid is a hexagon, this means that the cross-section is also a hexagon because it will be the same shape as its face. The correct option is C: A hexagon.

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Example 4 Which two of the objects below could have the following cross-section?

A Triangular pyramid

Cylinder

C Pentagonal pyramid

Pentagonal prism

Create a strategy

Apply the idea

Any solid with a pentagonal base should have a pentagonal cross-section.

The correct options are C and D because they have a pentagonal base.

Idea summary Prisms have rectangular sides and the shape on the top and the base is the same. Any cross-section parallel to the base is always the same and the shape of the cross-section is what gives the solid its name. Pyramids have triangular sides. Any cross-section parallel to the base is always the same shape, but is smaller in size than the base.

13.01 Practice questions Understanding 1

Adventurer 1, 2

Trailblazer 1, 2

Name these solids: a

Explorer 1

Explain what the net of a solid is.

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Fluency 3

Ex 2

Name the 3D shapes formed by these nets: b

A rectangular prism has been sliced through a plane parallel to its base. What two-dimensional shape is formed by the slice?

518

Name the solid that can be formed from the given sets of 2D shapes: a

Ex 3

3–8

Draw a net for these solids:

3–8

Name the solid formed from these nets: a

Ex 1

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Ex 4

A solid is cut parallel to one of its faces to give this cross-section: What could the solid possibly be?

Reasoning

9, 10

9–11

Describe how to use the net of a solid to find the solid’s surface area.

Explain the similarities and differences between a prism and a pyramid.

Jack wants to make a cardboard box in the shape of a rectangular prism with dimensions 10 cm × 8 cm × 6 cm. Sketch a net that Jack could use to make the box without using a measuring tool. Explain your reasoning.

Problem-solving 12 12

12, 13

Draw a net for each of the following items: a

Cereal box

Tissue box

Camping tent

6-sided die

Identify the relationship between the number of faces in a net and the number of edges and vertices in its corresponding 3-dimensional solid.

13.02 Visualise solids After this lesson, you will be able to… • identify and draw different views of solids and understand how they can be useful in various fields. • visualise 3D shapes from aerial views and footprints of structures.

Views of isometric drawings and interpreting perspective An isometric drawing refers to a 2D representation of a 3D object. This type of drawing has various views. Three-dimensional objects can be represented with the side view (the view on the right side of the object specifically), front view and top view clearly indicated on a two-dimensional surface. We can then ask about the view from each of these elevations.

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Consider this solid formed from cubes: Top

Front view

Side view

What is the side view? We can colour the sides of the cubes that are facing the side to make an image like this.

We can now piece together the side view by joining the highlighted faces together.

Side view

We can do the same from above to get the top view.

Looking up and over from the side, we can tell that this is the top view.

Top view

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When looking at the front view, we can see that this time there is a highlighted face that would be hidden from the view and this is not included.

Hidden

Once we have identified this hidden side, we can draw the front view.

Front view

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Another way to represent a 3D object on a 2D surface while showing its depth and structure is with a perspective drawing. By interpreting these drawings along with isometric drawings, we can visualise the 3D object and understand its properties.

Isometric drawing

Perspective drawing

In an isometric drawing, the object looks like it’s equally sized in all directions. It doesn’t have that realistic look, where things closer to us appear bigger and things farther away appear smaller. It’s like looking at a 3D object in a video game, where everything is even and lines stay parallel.

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On the other hand, perspective drawings try to show objects the way we see them in real life. Things that are closer to us look bigger and things that are farther away look smaller. In perspective drawings, lines that are parallel in real life might look like they’re coming together at a point. This helps create the feeling of depth, making the drawing look more realistic.

Example 1 Consider this solid formed from cubes:

Top

Front view

Side view

a Which of these diagrams represents the top view? A

Create a strategy

Apply the idea

Try imagining the tops of each cube as a different colour:

Joining the highlighted faces together, the diagram which represents the top view is:

The correct answer is Option B.

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b Which of these diagrams represents the front view? A

Create a strategy

Apply the idea

Try imagining the sides of each cube as a different colour:

Joining the highlighted sides of each cube, the diagram which represents the front view is:

The correct answer is Option B.

c Which of these diagrams represents the side view? A

Create a strategy

Apply the idea

Try imagining the fronts of each cube as a different colour:

Joining the highlighted fronts of each cube, the diagram which represents the side view is:

The correct answer is Option D.

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Idea summary Three-dimensional objects can be represented with the side view, front view and top view: Top

Front view

Side view

Different views of solids When looking at a prism, we can look at the prism from a 3D view or from one of the 2D views. A 3D view of the prism shows us what the whole solid looks like from an angle. From this perspective, we can see the faces that will be visible when looked at directly from one of the 2D views. The 2D views show what the solid looks like when viewed directly from the front, side or top. In the 3D view of this rectangular prism, we can see the sides that will be visible in the 2D views. In each 2D view, we look at the rectangular prism directly from either the front, side or top. Top view

Side view Front view 3D view

Front view

Side view

Top view

As we can see from the images, the front, side and top views are all 2D shapes that match the faces of the prism visible from each view when looked at directly. The plan view is another name for the top view and is used in architecture when referring to the plan of a building, which is how the building looks when viewed from directly above. Since the 2D views of a solid only show what the solid looks like directly from one angle, these views cannot show depth like a 3D view can.

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For example, if we look at this hexagonal prism from the front, we will be able to see these three faces.

However, two of these faces are sloped and will appear thinner in the front view than they actually are. As a result, the front view will look like this.

A 2D view might also show a side that we can’t see from the 3D view. Top view

Front view

When looking at a triangular prism from the 3D view, we notice that we can only see two of the five faces. The bottom and back faces of the prism won’t show up on any of our 2D views, but the last hidden face will be visible from the top view.

Side view

We find that the top view of this triangular prism looks like this. The fact that we can see two faces from the top view is shown by the line dividing the view into two rectangular faces. Aerial views of buildings and structures can help us visualise their footprints and identify the 3D shapes that could approximate their structure. This is useful in architecture, design and urban planning. The footprint of a building is the shape of its base on the ground.

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Example 2 Choose the solid that would have this front view:

A Top view

Top view

Side view

Front view

C Top view

Side view

Side view Top view

Front view

Create a strategy Look at the shape at the front of each solid.

Apply the idea We need to choose the solid that has the rectangle at its front. The rectangular prism has the rectangle at its front, so the answer is option D.

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Side view

Example 3 Consider the different views of this trapezoidal prism from different angles.

Top view

Front view

Side view

a What is the front view? A

Create a strategy Think of the 2D shape that we would see if we looked directly at the front of the given shape.

Apply the idea We can see a trapezium in front, so the answer is option A.

b What is the side view? A

Create a strategy Think of the 2D shape that we would see if we looked directly at the side of the given shape.

Apply the idea We can see a vertical rectangle to the side, so the answer is option B.

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c What is the top view? A

Create a strategy Think of the 2D shape that we would see if we looked directly at the top of the given shape.

Apply the idea The top view would show a rectangle. In the top view, you would see the length and width of the prism, which are parallel to the bases of the trapezium faces, forming a horizontal rectangle. So the answer is option C.

Idea summary When looking at a prism, we can look at the prism from a 3D view or from one of the 2D views. The 2D views show what the solid looks like when viewed directly from the front, side or top. Top

Side view Front view 3D view

Front view

Side view

Top view

Since the 2D views of a solid only show what the solid looks like directly from one angle, these views cannot show depth like a 3D view can. 2D views can’t show that one end is further away than the other or that one side of a solid is curved when we look directly at it. Aerial views help us visualise the footprints of buildings and structures, which can guide us in identifying 3D shapes that approximate their structure.

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13.02 Practice questions Understanding

Adventurer

1, 2

Trailblazer 1, 2

What is an isometric drawing?

Which type of drawing is more realistic, an isometric drawing or a perspective drawing?

Fluency Ex 1

Explorer

3a-c, 4a-c, 5, 6a-c, 7–11, 12a-d

3c-e, 6c-e, 9, 12c-f

3de, 4b-d, 6d-f, 9, 12g-j

For each solid formed from cubes: i

Draw the diagram for the front view.

iii

Draw the diagram for the top view.

Top view

Front view

Draw the diagram for the side view.

Top view

Side view

Side view Front view

Top view

Front view

Side view

Top view

Front view

Side view

Top view

Front view

Side view

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Select the solid that matches the given views. The side view is on the right of the shape. a

Front view

Side view

Top view

Top

Side view

Front view

Top

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Front view

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Side view

Front view

Which of these diagrams shows the top view of this collection of cylinders?

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Name the solid that would have these views: a

Front view

Side view

Top view

Front view

Side view

Top view

Front view

Side view

Top view

Front view

Side view

Top view

Front view

Side view

Top view

Front view

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Side view

Top view

Name the solid that would have this shape as one of its views: a

Ex 2

Choose the solid that has this rectangle as its front view:

Top view

Side view

Side view Front view

Front view

Top view

Side view Front view

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Choose the solid that has this rectangle as its side view: A

Top view

Side view Side view Front view

Front view

Top view

Side view

Front view

Side view

Choose the solid that has this as its top view:

Top view

Front view

Top view

Side view

Side view Front view

Top view

Side view

Front view Front view

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Choose the solid that has this as its top view:

Top view

Side view Side view

Front view Front view

Top view

Side view Side view Front view

Front view

Did you know?

Aerial views help architects design functional, well-balanced buildings by showing how they fit within city blocks, roads, and footpaths.

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Ex 3

For each solid: i

Sketch the front view.

Top view

Sketch the side view.

iii

Sketch the top view.

Top view

Front view

Side view

Front view

Top view

Side view

Front view

Side view Front view

Top view

Side view Side view

Front view Front view

Top view

Side view

Side view Front view

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Front view

Top view

Side view

Front view Side view

Front view

Reasoning 13

13, 14

13, 15

Choose the solid that has this as its top view and explain your reasoning:

Top view

Side view Side view Front view

Front view

C Top view

Top view

Side view Front view

Front view

Side view

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Yashi is studying visualising solids. Yashi states that these solids have the same front view, but different side and top views. Explain Yashi’s thinking.

For this solid, explain why a side view alone is an insufficient visualisation.

Problem-solving 16

16, 17

Choose a real-world object. Draw the diagrams for its side view, front view and top view.

Draw any solid, oriented in two different ways.

13.03 Volume using unit cubes After this lesson, you will be able to… • calculate the volume of rectangular prisms by using the concept of counting the number of rows of similar layers.

Volume using unit cubes The unit cube

1 unit3 of volume

1 unit 1 unit 1 unit

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The space inside a 3D shape is called its volume. We measure volume by imagining how many unit cubes, each with sides that are 1 unit long, can fit inside the shape. One of these small cubes has a volume of 1 cubic unit, written as 1 unit3.

Interactive exploration

Discover this concept in action online

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There is a relationship between the edge lengths of a rectangular prism and the prism’s volume. We can count 2 units the number of unit cubes that make up a prism, but we have a quicker way to find the volume.

3 units 5 units

This image shows a rectangular prism with length 5 units, width 3 units and height 2 units. Pay attention to the length of each side of the shape. The length shows how many unit cubes can be placed in a row along that side.

2 unit cubes

To figure out how many smaller cubes can fit inside a rectangular prism, we can count all the unit cubes one by one. Notice that this total will be equivalent to multiplying the three side lengths: 5 × 3 × 2 = 30

5 unit cubes

3 unit cubes

so there are 30 unit cubes in the prism, which means it has a volume of 30 unit3.

Example 1 In this figure, 1 block = 1 cm3. What is the volume of the solid figure?

Create a strategy

Apply the idea

Count the unit cubes one by one or multiply the 3 edge lengths of the rectangular prism.

We can see on the figure that the front face is an array of blocks with 4 rows and 3 columns. We have 2 sets of this face. Volume = 3 × 4 × 2 Multiply the columns, rows and sets = 12 × 2

Evaluate 3 × 4

= 24 cm3

Evaluate 12 × 2

Idea summary Volume means the amount of space a 3D shape takes up. To find the volume, we can count the number of unit cubes that fit inside the shape.

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13.03 Practice questions Understanding 1

Volume

20 unit cubes

540

4 unit cubes

25 unit cubes

There are 2 layers and in each layer, there are 9 cubes.

There is 1 layer and in each layer, there are 16 cubes.

There are 6 layers and in each layer, there are 6 cubes.

There are 11 layers and in each layer, there are 9 cubes.

There are 14 layers and in each layer, there are 10 cubes.

16 unit cubes

Identify how many different rectangular prisms can be made using the numbers of cubes given and provide their dimensions. 12 cubes

Fluency 5

Unit cube

Identify the total number of unit cubes for these statements:

Ex 1

Trailblazer 1, 2, 3c-e, 4cd

Sketch a rectangular prism that is made up of: a

Adventurer 1, 2, 3c-e, 4ab

Explain what these terms mean: a

Explorer 1, 2, 3a-c

48 cubes

72 cubes

In these figures, 1 block = 1 cm3. Find the volume of each solid figure. a

Mathspace WA Curriculum – Year 7 mathspace.co

120 cubes

Reasoning

7, 8

Will the volume of a rectangular prism change if it is turned on its side, changing its length, width and height? How do you know?

Explain how to use 1 centimetre cubes to fill a 1 metre cube-shaped cardboard box.

How can we use unit cubes and a rectangular prism to explain how the formula for the volume of a rectangular prism, V = l × w × h, works?

Problem-solving 9 9

Imagine you have a large cube that is 5 units long on each side and made up of smaller cubes, each with a side length of 1 unit. If you remove all the inner cubes to only leave the shell, what is the volume of the remaining structure?

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13.04 Units of volume After this lesson, you will be able to… • estimate volume and capacity. • identify appropriate units for measuring volume and capacity.

Units of volume Volume is how much space something takes up, like a 3D shape or even a solid, liquid or gas. Capacity is different - it’s about how much a container can hold, like how much juice can fit in your cup. We usually talk about capacity when we’re dealing with liquids in containers. Common units for volume from smallest to largest are: • cubic millimetres (mm3) • cubic centimetres (cm3) • cubic metres (m3) Common units for capacity from smallest to largest are: • millilitres (mL) • litres (L) • kilolitres (kL) • megalitres (ML) Just as with lengths and areas, to determine the correct unit to use and to be able to estimate the volume or capacity of an object, it is useful to identify some common items that are about the size of the units above. Such as: • 1 cubic millimetre, 1 mm3, is very small. This is a cube with length, width and height 1 mm — about the size of 10 grains of salt. • 1 cubic centimetre, 1 cm3, is the size of a cube with length, width and height 1 cm — this is about the size of a die. • 1 cubic metre, 1 m3, is the size of a cube with length, width and height 1 m — about the size of a fridge. • 1 millilitre, 1 mL, is the same as 1 cm3, and a teaspoon is 5 mL. • 1 litre, 1 L. It is very common for milk to come in 1 and 2 litre cartons. • 1 kilolitre, 1 kL, is one thousand litres. This is equal to 1 m3. • 1 megalitre, 1 ML, is one million litres — this is a lot of liquid. An Olympic swimming pool holds on average 2.5 ML.

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Example 1 When referring to the capacity of a pool, which of these units could be an appropriate unit to use? Select all correct options. A mL

B L

C ML

Create a strategy Think about possible sizes of a pool.

Apply the idea If we are looking at an olympic size swimming pool, which is very large, then we could use megalitres to measure the capacity. If we looking at a kid’s pool or a home pool, that’s a lot smaller, we could probably use litres to measure the capacity. The answers are options B and C.

Example 2 Which of these options are usually measured in cubic centimetres? A The volume of air in a hall B The volume of a bottle C The area of a sheet of paper D The amount of space taken up by a textbook

Create a strategy Choose which among the choices would be an appropriate size to be measured in cubic centimetres. Use the fact that one cubic centimetre is about the volume of a die.

Apply the idea Option A is not appropriate to measure in cubic centimetres as the hall is too big for this unit of measurement of volume. Option C is not measured using volume as it is area. Options B and D will be measured in cubic centimetres because a bottle and the amount of space taken up by a textbook are of appropriate sizes.

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Example 3 What is the best estimate for the volume of liquid in this container if it is able to hold 3600 millilitres when full? A 2200 millilitres

1800 millilitres

C 1200 millilitres

3200 millilitres

Create a strategy Compare the part of the container that has been filled with water against the part that is empty.

Apply the idea Notice that the water line is around way up the height of the container. However, the part of the container that has been filled with water is much wider than the part that is empty. This means that over of the container is filled with water. Options B and C are not good enough estimates for the volume of liquid since more than container is filled with water and these estimates are too low.

of the

Option D is also not a good enough estimate for the volume of liquid as this estimate seems too much compared with the container filled with more than half of the water. So, Option A is a good enough estimate for the volume of liquid in the container.

Idea summary Volume is the amount of space an object takes up. This can be the amount of space a 3D shape occupies or the space that a substance (solid, liquid or gas) fills. Common units for volume from smallest to largest are: • cubic millimetres (mm3) • cubic centimetres (cm3) • cubic metres (m3) Capacity is how much a container can hold, like how much juice can fit in your cup. We usually talk about capacity when we’re dealing with liquids in containers. Common units for capacity from smallest to largest are: • • • •

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millilitres (mL) litres (L) kilolitres (kL) megalitres (ML)

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13.04 Practice questions Understanding

Trailblazer 1, 2

Explain the difference between volume and capacity.

Give examples of real-world objects that could have the given volume or capacity: a

1 mm3

1 cm3

1 m3

1 kL

1 ML

3–8

7–12

6–12

Identify the most appropriate unit for measuring the volume of: a

Sand pit

Wardrobe

Glass of water

Which units can we measure with a jug like the one in the picture?

Identify the most appropriate unit for measuring the capacity of a coffee cup.

Identify the most appropriate unit for measuring the capacity of a fish bowl.

A landscaper needs to move soil across the garden. Choose the best estimate for the volume of soil that can be carried by a wheelbarrow in one trip. A

1 mL

Determine whether these units are appropriate to use to measure the water needed to sustain the population of a small city. a

Ex 2

Adventurer 1, 2

Fluency Ex 1

Explorer 1, 2a-e

771 mL

21 ML

Air in a hall

71 L

Consider this matchbox: a

Identify the most appropriate unit for measuring the volume of a matchbox.

Estimate the volume of the matchbox.

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Ex 3

This suitcase has a total capacity of 88 500 cm3. Estimate the volume of the shirts contained in the suitcase.

Estimate the volume of liquid in this container if it is able to hold 3.6 L when full.

Estimate the volume of liquid in this container if it is able to hold 3.8 litres when full.

Reasoning

13, 14

Estimate the volume of liquid in this container if it is able to hold 2400 mL when full. Explain the steps you took to decide.

The volume of the bookshelf shown is 0.64 m3. Estimate the volume of books on the bookshelf and explain how you calculated your estimation: a

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Problem-solving 16

15–17

A town’s reservoir contains 10 million megalitres of water and the daily demand for water is 50 000 megalitres. If the demand remains constant, how many days will the reservoir supply the town’s water needs without further water supply?

A full jug of milk has a total capacity of 1.5 litres. If 300 mL of milk is used to make a cake, how much milk is left in the jug?

A 500 mL bottle of soda is sold for $2.50, while a 2 L bottle of the same soda is sold for $7.80. a

Which is a better value buy?

What is the cheapest way to buy 3 L of this soda?

13.05 Volume of right prisms After this lesson, you will be able to… • find the volume of a variety of right prisms and composite solids using appropriate formulas and methods.

Volume of a rectangular prism Interactive exploration

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Since the area of the cross-section parallel to its base is the same size throughout a rectangular prism, we find the area of the base and multiply by the number of rows or the height of the prism. The volume of a rectangular prism is given by: Volume = area of base × height or

Heigth

Width Length

Volume = length × width × height or V = l × w × h

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A cube is a special type of rectangular prism, one that has all sides equal in length. The volume of a cube is given by: Volume = area of base × height or

Side

Volume = side × side × side or

Side

V = s × s × s = s3

Side

In general, the formula for the volume of a right rectangular prism or a cube is given by V = Abase × h where Abase is the area of the prism’s base and h is the height of the prism.

Example 1 Find the volume of the rectangular prism: 12 cm

5 cm 7 cm

Create a strategy Calculate the area of the base of the rectangular prism, then multiply that by the height.

Apply the idea Based on the diagram above, we are given l = 12, w = 7 and h = 5. A=l×w

Write the formula for the area of the base

= 12 × 7 = 84 cm

Substitute the values 2

Evaluate

Now, when we multiply the area of the base by the height of the rectangular prism, we have the volume of the rectangular prism: V = Abase × h = 84 × 5 = 420 cm

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Write the formula Substitute the values

Evaluate

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Idea summary The volume of the rectangular prism is given by:

V=l×w×h V l w h

is the volume of the rectangular prism is the length of the rectangular prism is the width of the rectangular prism is the height of the rectangular prism

The volume of the cube is given by:

V = s × s × s = s3 V s

is the volume of the cube is the side length of the cube

In general, the formula for the volume of a right rectangular prism or a cube is given by V = Abase × h where Abase is the area of the prism’s base and h is the height of the prism.

Volume of other right prisms Interactive exploration

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For other types of right prisms, such as triangular prisms, the volume can be found by multiplying the area of the base, Abase, by the height of the prism, h. The formula for the volume of a right prism is V = Abase × h, which follows from the general formula for the volume of a rectangular prism.

Example 2 Consider the right prism:

5 cm

4 cm

23 cm

6 cm

a What is the shape of the prism’s base?

Create a strategy

Apply the idea

The base of a prism is the same as the cross-section of the prism.

The shape of the base is a triangle.

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b Calculate the area of the prism’s base.

Create a strategy Use the formula for the area of a triangle to calculate the area of the base.

Apply the idea

Write the formula

Substitute the values

Evaluate

The area of the prism’s base is 12 cm .

c Calculate the volume of the prism.

Create a strategy The volume of a right prism is calculated by multiplying the area of its base by its height.

Apply the idea V = Abase × h

Write the formula Substitute Abase = 12, h = 23

= 12 × 23 3

= 276 cm

Evaluate the multiplication

The volume of the triangular prism is 276 cm3.

Idea summary The formula for the volume of a rectangular prism, the volume of a triangular prism and other right prisms is calculated with the formula V = Abase × h where Abase is the area of the prism’s base and h is the height of the prism.

Volume of composite solids If a composite solid is made up of simpler solids or prisms, then the volume of the whole solid is just the total of the volume of all the smaller solids put together. This complicated shape is built from simple rectangular prisms:

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Example 3 Find the volume of the composite solid by breaking it up into rectangular prisms. 12 cm 3 cm

3 cm 7 cm

3 cm

Create a strategy 12 cm 3 cm

3 cm

Break up the composite solid into two smaller rectangular prisms. The volume of the entire shape is found by adding together the volume of the two smaller rectangular prisms.

7 cm 3 cm 3 cm

Apply the idea Vcomposite = Vtop prism + Vbottom prism = (12 × 3 × 3) + (3 × 3 × 7)

Add the two smaller volumes Substitute the values

= 108 + 63 Evaluate the multiplications Evaluate = 171 cm3

Idea summary The volume of a composite solid is the sum of the volume of each solid it is built from.

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13.05 Practice questions Explorer

Understanding

Adventurer

Trailblazer

Describe how the volume of a composite solid could be calculated.

Explain why every cube a rectangular prism but not every rectangular prism is a cube.

Fluency 3 Ex 1

3–7, 9–11

3cd, 5, 6, 8, 10–12

3cd, 5, 6, 8, 11–13

Find the volume of these prisms: a

9 mm

10 mm

8 mm

50 cm

9 cm

5 cm

9.9 m 12 m 4m

8.7 m 7.5 m

4 cm 2 cm

11 cm 8 cm 4 cm

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4 cm

Find the volume of these cubes: a

6 cm 12 cm

Find the volume of a cube that has a side length of 14 m.

Find the side length of a cube that has a volume of 27 cm3.

A container is a rectangular prism with dimensions 50 cm, 30 cm and 60 cm. Find the volume of the container.

A regular box of cereal is a rectangular prism and measures 25 cm by 7 cm by 28 cm. a

Find the volume of the box of cereal.

The company that makes these cereal boxes also makes a jumbo size box, which is one and a half times as long, wide and tall as the regular size boxes. i

Find the volume of the jumbo box of cereal.

Is the volume of the jumbo box one and a half times the volume of the regular box?

A coffee table has the shape of a triangular prism with dimensions of a base area of 980 cm2 and a height of 70 cm. Find the volume of the table.

A box is shaped like a triangular prism, with a base length of 50 cm and a height of 30 cm for its triangular face. The height of the prism is 60 cm. Find the volume of the box.

Ex 2

The composite solid can be broken down into two smaller rectangular prisms:

Ex 3

6 cm 6 cm

6 cm 7 cm

6 cm

14 cm

Find the volume of the top prism.

Find the volume of the bottom prism.

Then, find the total volume of the composite solid.

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The composite solid can be broken down into two smaller rectangular prisms: 2 cm

14 cm 2 cm 7 cm 8 cm

Find the volume of the top prism.

Find the volume of the bottom prism.

Then, find the total volume of the composite solid.

Find the volume of these composite solids: a

1 cm

13 cm

7 16

4 cm 10 cm

10 cm 2 cm

d 2 cm

7 cm 2 cm 24 cm

2 cm

Reasoning 14

15, 16

14, 16

Find two different ways to calculate the total volume of the composite solid and explain your thinking. Determine which of the two methods could be considered more efficient and why. 7 cm

16 cm

A box is 2 m long, 30 cm high and 40 cm wide. Write the steps to calculate the volume of the box in cubic centimetres.

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This solid consists of a rectangular prism with a smaller rectangular prism cut out of it. a

Explain how to find the volume of the solid, then calculate the volume of the solid.

Is there another way the volume could be found? If so, explain.

Problem-solving 17, 18

12 cm 4 cm

7 cm

6 cm

17–20

18 cm

18–21

A tank has a length of 9 m, width of 4 m and depth of 8 m. Find the volume in cubic metres and then use that to find the capacity of the tank in litres.

Consider the aquarium: a

Calculate the maximum volume of water that the aquarium can hold and use that to find the total capacity in litres.

Explain what reasonable volume of water would be needed to fill the aquarium and how you determined your answer.

50 cm 30 cm

25 cm

A rectangular prism has a volume of 120 cubic units. It is known that the length, width and height are all whole numbers. If the length and width of the prism are in a ratio of 3: 4, what is the height of the rectangular prism?

A juice box is in the shape of a rectangular prism with a length of 8 cm, a width of 4 cm and a height of 12 cm. If the juice box is filled with 250 mL of juice, what is the height of the juice in the box?

Hanni is constructing a tent in the shape of a triangular prism in the woods. The base of the tent measures 2.4 m and its height is 3.4 m, while the total length of the tent is 4 m. If Hanni estimates that her friends will occupy space in the tent?

of the tent, what is the volume of the remaining

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13.06 Attributes in measurement formulas After this lesson, you will be able to… • manipulate the area formula, formula for volume of a rectangular prism and formula for speed, distance and time to investigate the effect of changing variables. • produce a table of values and draw a graph of the values to understand the linear and non-linear relationships between the variables.

Manipulate the area formula Interactive exploration

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Consider the formula for the area of a rectangle: Area = length × width

Length

or A=l×w Width

where l and w are variables.

When the length or width of a rectangle is a constant, we called that value fixed. The relationship between the area of a rectangle and the product of the length and width as one of the dimensions changes then becomes a linear relationship.

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Example 1 Consider these rectangles and their dimensions: 1 cm

2 cm

3 cm

A=1×3 A = 3 cm2

3 cm

A=2×3 A = 6 cm2

A=3×3 A = 9 cm2

a For the rectangle above, with a fixed width of 3 cm, complete the table for various lengths. Width Length Area

3 1

3 2

3 3

3 4

3 5

3 6

3 7

3 8

3 9

3 10

Create a strategy We can use the formula A = l × w to find the area of the rectangle.

Apply the idea Width Length Area

3 1 3

3 2 6

3 3 9

3 4 12

3 5 15

3 6 18

3 7 21

3 8 24

3 9 27

3 10 30

b Now consider the length fixed as 5 cm. Complete the table for various widths. Width Length Area

Create a strategy Multiply the length and width, given the constant length of 5 cm.

Apply the idea Width Length Area

1 5 5

2 5 10

3 5 15

4 5 20

5 5 25

6 5 30

7 5 35

8 5 40

9 5 45

10 5 50

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c Graph the functions for area varying the length and varying the width on the axes shown, labelling each carefully.

Area

45 40 35 30 25 20 15 10 5 1

Create a strategy Plot the points on a Cartesian plane by using the values from the table. Use the value of the length as the x-coordinate and the corresponding area as the y-coordinate to plot each point. Connect the points to form a line or a curve.

Length / width 7 8 9

Apply the idea 50

Area

45 varying width length = 5 cm

40 35 30 25 20

varying length width = 3 cm

15 10 5

Length / width 1

2 3 4 5 6 7 8 9 10

d Determine the formula that can be used to calculate the area of a rectangle with a fixed width of 3 cm.

Create a strategy

Apply the idea

Use the formula A = l × w. Substitute the value of the constant dimension.

For w = 3, A = l × 3 or A = 3l

e Determine the formula that can be used to calculate the area of a rectangle with a fixed length of 5 cm.

Create a strategy

Apply the idea

Use the formula A = l × w. Substitute the value of the constant dimension.

For l = 5, A = 5 × w or A = 5w

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Idea summary We can investigate the effect of changing the variables in a formula like A=l×w by fixing one of the variables and seeing what happens when we vary the other by: • Producing a table of values • Drawing a graph of the values he graphs produced in this case were straight lines so we say there is a linear T relationship.

Manipulate the formula for volume of a rectangular prism Since the area of the cross-section parallel to its base is the same size throughout a rectangular prism, we find the area of the base and multiply by how many rows, or the height, that we have. Each side is at a right angle (like an “L”) to the other two sides. The volume of a rectangular prism is given by: Volume = area of base × height or Height

Volume = length × width × height

Width

Length

V = l × w × h When the length, width or height of a rectangular prism is a constant, we called that value fixed. The relationship between the volume of a rectangular prism and the product of the length, width and height as one of the dimensions changes then becomes a linear relationship.

Example 2 Consider the volume of the rectangular prisms. They all have a height of 2 cm and a width of 3 cm. l = 1 cm

l = 2 cm

l = 3 cm

w = 3 cm

h = 2 cm

a What is the volume of the block with length 1 cm?

Create a strategy Use the formula V = l × w × h to find the volume of the block.

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Apply the idea V=1×3×2

Substitute the values for length, width and height

Evaluate

The volume is 6 cm3.

b What is the volume of the block with length 2 cm?

Create a strategy Use the formula V = l × w × h to find the volume of the block.

Apply the idea V=2×3×2

Substitute the values for length, width and height

= 12

Evaluate 3

The volume is 12 cm .

c What is the volume of the block with length 3 cm?

Create a strategy Use the formula V = l × w × h to find the volume of the block.

Apply the idea V=3×3×2

Substitute the values for length, width and height

= 18

Evaluate 3

The volume is 18 cm . d Complete the table: Length Volume

Create a strategy To find the volume of a rectangular solid, use the formula V = l × w × h. Keep in mind that the width (w) and height (h) values are constant.

Apply the idea Since the width and height are both 2 and 3, respectively, we can calculate the volume by multiplying the length (l) by 6 to complete the table. Length Volume

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1 6

2 12

3 18

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5 30

6 36

7 42

8 48

9 54

10 60

e Graph the relationship between the length of these blocks and their volume on the axes.

Volume (cm)3

50 45 40 35 30 25 20 15 10 5

Length (cm) 1

Create a strategy Plot the points on a Cartesian plane by using the values from the table. Use the value of the length as the x-coordinate and the corresponding volume as the y-coordinate to plot each point. Connect the points to form a line or a curve.

Apply the idea 50

Volume (cm)3

45 40 35 30 25 20 15 10 5

Length (cm) 1

Find a formula relating the length and the volume in the form V = kl

Create a strategy Since we know that the height (h) is 2 and the width (w) is 3, we can substitute these values into the formula.

Apply the idea V=l×3×2

Substitute the values of width and height

V = 6l

Evaluate

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Example 3 Consider the rectangular prism. l = 1 cm w = 3 cm

h = 1 cm

a What is the volume of this rectangular prism?

Create a strategy Use the formula V = l × w × h.

Apply the idea Let l = 1, w = 3 and h = 1: V=1×3×1

Substitute the values of length, width and height

Evaluate

The volume of the rectangular prism is 3 cm3.

b Suppose that we are going to increase the volume by adding extra blocks to increase the height while keeping the width and length the same. Select the diagram that shows the next rectangular prism in the sequence. A

Create a strategy To increase its height, we would add more identical prisms of dimensions 1 unit long, 3 units wide and stack them on top of the existing prism.

Apply the idea Add more identical rectangular prisms (or blocks) of the same dimensions to the existing prism to increase its height. The correct diagram is C.

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c Complete the table: Height Volume

Create a strategy Use the formula for the volume of a rectangular prism: V = l × w × h. Given that the length l = 1 cm and width w = 3 cm are constant, we will multiply the height of the prism by 1 × 3 to get the volume and complete the table.

Apply the idea Height Volume

1 3

2 6

3 9

4 12

5 15

6 18

7 21

8 24

9 27

10 30

d Find a formula relating the height and the volume of the form V = kh.

Create a strategy

Apply the idea

Given the length (l) is 1 cm and the width (w) is 3 cm, we can substitute these values into the formula.

V = 1 × 3 × h Substitute the values of length and width = 3h

Evaluate

e What would be the volume of a prism like the ones above if the height was 20 cm?

Create a strategy Substitute the value of the height to the formula V = 3h.

Apply the idea V = 3(20) = 60

Substitute the value of h Evaluate

The volume is 60 cm3.

Idea summary We can investigate the effect of changing the variables in a formula like V=l×w×h by fixing two of the variables and seeing what happens when we vary the other by: • Producing a table of values • Drawing a graph of the values The graphs produced in this case were straight lines so we say there is a linear relationship. We can extend this idea to formulas with lots of variables.

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Manipulate the formula for speed, distance and time Speed, distance and time are important concepts in understanding how things move. They are connected by the formula: Speed = Distance ÷ Time or • Speed (S) tells us how fast something is moving. It can be measured in different units, like kilometres per hour (km/h) or metres per second (m/s). • Distance (D) is the space between two points and is measured in units like kilometres or metres. • Time (T ) is how long it takes to travel that distance and is usually measured in hours, minutes or seconds. Using this formula, we can find the speed of any object if we know the distance it travelled and the time it took to travel that distance.

Example 4 Consider a journey of 720 km. a Calculate the average speed of the journey for each time given in the table: Time (hours) Speed (km/hr)

Create a strategy Use the formula

Apply the idea Divide the distance D = 720 by the time T to complete the table. Time (hours) Speed (km/hr)

1 720

2 360

3 240

4 180

6 120

b Graph the relationship between the time taken and the average speed on the axes shown.

8 90

700 650 600 550 500 450 400 350 300 250 200 150 100 50

9 80

Speed (km / hr)

Time (hrs) 1

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10 72

9 10

Create a strategy

Apply the idea

Plot the points on a Cartesian plane by using the values from the table. Use the value of the time as the x-coordinate and the corresponding speed as the y-coordinate to plot each point. Connect the points to form a line or a curve.

700 650 600 550 500 450 400 350 300 250 200 150 100 50

Speed (km / hr)

Time (hrs) 1

9 10

c Is this relationship linear?

Create a strategy

Apply the idea

Observe the graph and check if it appears as a straight line.

The graph is a curve, so it is not linear.

Idea summary We can investigate the effect of changing the variables in a formula like

by fixing one of the variables and seeing what happens when we vary the other by: • Producing a table of values • Drawing a graph of the values ot all relationships are linear. We can see the shapes of different relationships by N graphing.

13.06 Practice questions Understanding 1

Explorer 1–3

Adventurer 1–3

Trailblazer 1, 3

Identify these variables as constant or varying: a

Length is 7 cm

Width is w cm

Height is H m

Distance is 45 km

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Consider the table with the variables of distance, time and speed. Distance (km) Time (hours) Speed (km/h)

120 1 120

120 3 40

120 4 30

120 5 24

120 10 12

Which variable is constant in this table?

What is the value of the constant?

Which variable(s) is increasing?

Which variable(s) is decreasing?

The table displays the values when calculating the area of a particular right angled triangle. 12 1 6

12 2 12

12 3 18

12 4 24

12 5 30

12 6 36

12 7 42

12 8 48

12 9 52

12 10 60

Name the three variables.

Which variable is constant in this table?

What is the value of the constant?

Which variable(s) is increasing?

Which variable(s) is decreasing?

Complete the rule for the area of this triangle, where the base is 12 cm: Area =

× height

Fluency

4, 5, 7–9

5–7, 9

A rectangle has a fixed area of 32 square centimetres. Use the table to list all the possible dimensions of the rectangle, if the length and width are whole numbers.

Consider a rectangle which has a fixed length of 4 cm and a variable width. a

Length (cm) Width (cm)

Complete the table for various widths: Length Width Area

4 1

4 2

4 3

4 4

4 5

4 6

4 7

4 8

4 9

4 10

From the table, write a rule that can be used to find the area of a rectangle with a fixed length of 4 cm.

Plot the points from the second and third row of the table and join them to graph the function for the area of this rectangle on the axes shown:

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120 6 20

Base (cm) Height (cm) Area

Ex 1

120 2 60

Is this function linear or non-linear?

Mathspace WA Curriculum – Year 7 mathspace.co

60 55 50 45 40 35 30 25 20 15 10 5

Area

Width 1

2 3 4 5 6 7 8 9 10

Consider a rectangle which has a fixed width of 6 cm and a variable length. a

Complete the table for various lengths: Length Area

From the table, write a rule that can be used to find the area of a rectangle with a fixed width of 6 cm.

Plot the points from the table and join them, to graph the function for area with fixed width and varying length on the axes shown:

d Ex 2

60 55 50 45 40 35 30 25 20 15 10 5

Area

Is this function linear or non-linear?

Consider the volume of the rectangular prisms. They all have a length of 4 m and a width of 2 m. l=4m

w=2m

l=4m

w=2m

Find the volume of the block with a height of 1 m.

Find the volume of the block with a height of 2 m.

Find the volume of the block with a height of 3 m.

Complete the table: Height Volume

w=2m

Graph the relationship between the height of these blocks and their volume on the axes shown.

Find a formula relating the height and the volume in the form V = kh.

80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5

Volume (m)3

Height (m) 1

l=4m

h=3m

h=2m

h=1m

Ex 3

Length 2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8 9 10

Consider a rectangular prism which has a length of 5 cm, a width of 2 cm and a height of 3 cm. a

Find the volume of the rectangular prism.

Suppose we are going to increase the volume by adding extra blocks to increase the height while keeping the width and length constant.

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Complete the table for various heights: Width Length Height Volume

2 5 4

2 5 5

2 5 6

2 5 7

2 5 8

2 5 9

2 5 10

Find a formula for this rectangular prism, relating the height and the volume in the form V = kh, where k is a constant.

Find the volume of this rectangular prism if the height is 35 cm.

A car travels a fixed distance of 360 km The average speed of the car is given by the rule:

Ex 4

Find the average speed of the car for each time given in the table: Distance (km) Time (hours) Speed (km/h)

360 1

360 2

360 3

360 4

Use the second and third row of the table of values to graph the relationship between time and speed for the fixed distance on the axes shown:

Is this relationship linear?

Find the speed of the car when the time taken is 15 hours.

360 5

60 55 50 45 40 35 30 25 20 15 10 5

360 6

10, 11

360 9

360 10

Speed (km / h)

Time (hours) 1

Reasoning

360 8

2 3 4 5 6 7 8 9 10

10, 11

The number 5 is an example of a constant and w is an example of a variable. Why do you think the words constant and variable were chosen to represent these mathematical terms?

A square has a fixed area of 144 cm2. Is the side length of the square a constant or a variable? Explain your answer.

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Problem-solving 12, 13 12

12–14

12–16

Consider the formula to find the area of a rectangle: A = lw where A is the area, l is the length and w is the width. Determine whether these statements are true or false:

If the length is 4 cm and the width is 3 cm, then the area is 12 cm2.

If w = 3 and l increases, A also increases.

If l = 2 and w increases, A decreases.

If A = 5w, the width is a constant value of 5.

Consider the formula of the volume of a rectangular prism: V = lwh where V is the volume, l is the length, w is the width and h is the height. Determine whether these statements are true or false:

If the width and length are constant and h decreases, then V also decreases.

If h is constant and w and l increase, V decreases.

If V = 10lw, the height is a constant of 10.

If the volume is 24 cm3, width is 4 cm and the length is 3 cm, then the height is equal to 2 cm.

The formula to find the speed, S, over distance, D, travelled in the time taken, T, is

Determine whether these statements are true or false: a

If the speed is 9 m/s and the distance travelled is 81 m, then the time is equal to 9s.

If the distance is constant and T increases, S also increases.

If the time is constant and D decreases, S also decreases.

, the time is a constant with value of 15.

A triangle has a fixed height and varying base. If we want the area to double, by how much should we increase the base?

Peter is driving a fixed distance. If he wishes his time to be reduced by half, by how much should he increase his speed?

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13 Chapter review 1

Name these solids: a

Draw a net for these solids: a

Name the solid formed by each net: a

For the solid formed from cubes: i

Draw the diagram for the side view.

iii

Draw the diagram for the top view.

Top

Front view

Draw the diagram for the front view.

Top view

Side view Front view

How many different rectangular prisms can be made using: a

20 cubes

24 cubes

Identify the most appropriate unit for measuring the volume of: a

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Side view

Swimming pool

Refrigerator

Mathspace WA Curriculum – Year 7 mathspace.co

Coffee cup

Car’s trunk

Find the volume of these prisms: a

4 mm 7 mm

10 mm

14 cm 3 cm 5 cm

13 mm

d 8 cm

15 mm 7 cm 5 cm

11 mm

A rectangular prism container has dimensions 25 cm, 35 cm and 50 cm. Find the volume of the container.

A regular box of cereal is in the shape of a rectangular prism and measures 23 cm by 11 cm by 19 cm.

Find the volume of the box of cereal.

The company that makes these cereal boxes also makes a jumbo size box, which is 1.5 times larger in all dimensions. i

Find the volume of the jumbo box of cereal.

Is the volume of the jumbo box one and a half times the volume of the regular box?

Find the volume of these prisms: a

5.3 cm 10 cm

11.5 cm 2 cm 8 cm 4 cm

10 cm

5 mm 10 mm

18 cm

6 mm 6 cm

8 cm

Chapter 13 review mathspace.co

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The volume of the triangular prism shown is 287.5 m3:

Find the value of k, to one decimal place. 10 m

The volume of the triangular prism shown is 247.45 cm3: Find the value of y to one decimal place. 7 cm y cm 9.9 cm

Find the volume of these composite solids: a

11 cm

2 cm

2 cm 8 cm 16 cm 2 cm 4 cm 9 cm

d 5 cm 6 cm

11 cm 4 cm

16 cm

4 cm 13 cm

A swimming pool has a length of 12 m, width of 11 m and depth of 6 m. Find the volume of the swimming pool.

A rectangular prism has a volume of 168 cubic units. It is known that the length, width and height are all whole numbers. If the length and width of the prism are in a ratio of 2:3, what is the height of the rectangular prism?

A rectangular container has a length of 10 cm, a width of 6 cm and a height of 15 cm. If 450 mL of liquid is poured in, what is the height of the liquid in the container?

Identify these variables as constant or varying: a

572

Length is 9 cm

Width is k cm

Mathspace WA Curriculum – Year 7 mathspace.co

Height is H m

Distance is 25 km

Consider a rectangle which has a fixed length of 6 cm and a variable width. a

Complete the table below for various widths: Length Width Area

6 1

6 2

6 3

6 5

6 6

6 7

6 8

6 9

6 10

From the table, write a rule that can be used to find the area of a rectangle with a fixed length of 6 cm.

Plot the points from the second and third row of the table and join them to graph the function for area of this rectangle on the axes below: 60 55 50 45 40 35 30 25 20 15 10 5

Area

Width 1

d 19

6 4

2 3 4 5 6 7 8 9 10

Is this function linear or non-linear?

Consider a rectangular prism which has a length of 6 cm, a width of 3 cm and a height of 5 cm. a

Find the volume of the rectangular prism.

Suppose that we are going to increase the volume by adding extra blocks to increase the height while keeping the width and length constant. Complete the table below for various heights: Width Length Height Volume

3 6 6

3 6 7

3 6 8

3 6 9

3 6 10

3 6 11

3 6 12

Find a formula for this rectangular prism, relating the height and the volume in the form V = kh, where k is a constant.

Find the volume of this rectangular prism if the height is 25 cm.

Consider the formula to find the area of a rectangle: A = lw where A is the area, l is the length and w is the width. Determine whether these statements are true or false: a

If the length is 5 cm and the width is 3 cm, then the area is 15 cm2.

If w = 3 and l increases, A also increases.

If l = 5 and w increases, A decreases.

If A = 5w, the width is a constant value of 5. Chapter 13 review mathspace.co

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Big ideas • Rigid transformations involve translating (sliding), rotating (turning) or reflecting (flipping) an object, that can be combined in a sequence without changing the object’s shape or size. • Line symmetry and rotational symmetry are important concepts in geometry that help us identify properties of shapes and develop spatial reasoning and critical thinking skills.

Curriculum Outcomes • Plot coordinates on the Cartesian plane and explore, visualise, predict and determine image coordinates after translation or reflection across the axes or rotation about the origin • Analyse, represent and interpret problems in real-world situations involving perimeter and area of squares, rectangles, triangles and rectangular composite shapes, parallel lines, properties of triangles, transformations of points, views of rectangular prisms and rectangular composite objects, volume and/or Australian time zones

14 Transformations Chapter outline 14.01 Transformation of shapes 14.02 Translations 14.03 Reflections 14.04 Rotations 14.05 Multiple transformations Investigation: Tessellations 14.06 Line and rotational symmetry Chapter 14 review

576 582 591 599 610 622 631

In animation and video games, transformations are used to make objects and characters move realistically and interact with the environment.

14.01 Transformation of shapes After this lesson, you will be able to… • identify and define different types of transformations, including translations, reflections and rotations.

Transformations of shapes A transformation occurs when an object or point is moved, turned, flipped or changed in shape or size. We are going to consider a few types of transformations in which the shape and size of the image remain the same. These transformations are known as: • Translations - a transformation where all the points on the object slide the same distance in the same direction. • Reflections - a transformation where the object is flipped over, resulting in a mirror image. • Rotations - a transformation where an object is turned around a fixed point. One method of determining the type of transformation is to draw the object on tracing paper, then decide how to move the tracing paper to get the new object. • Translation - slide the tracing paper • Reflection - flip the tracing paper over • Rotation - turn or spin the tracing paper A shape or object may be transformed more than once. Think about whether the position or orientation of the shape has changed to decide whether it has been translated, reflected or rotated.

Example 1 Choose the picture that shows a translation. A

Create a strategy

Apply the idea

Choose the option where both objects are facing the same way.

Option B is the correct answer because the sock has been slid but has not flipped or turned.

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Mathspace WA Curriculum – Year 7 mathspace.co

Example 2 Choose the picture that shows a reflection. A

Create a strategy Choose the option that shows two objects that are mirror-images of each other.

Apply the idea Option C is the correct answer because the two kangaroos are mirror-images of each other.

Example 3 Choose the picture that shows a rotation. A

Create a strategy

Apply the idea

Choose the option that shows two objects where one object is turned.

Option C is the correct answer because the hammer on the right has been turned.

14.01 Transformation of shapes mathspace.co

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Idea summary A transformation is when an object or point is moved, turned, flipped or changed in shape or size. • A translation slides an object. • A reflection flips an object over, creating a mirror image. • A rotation spins an object or turns it around a fixed point.

14.01 Practice questions Understanding

Adventurer

Trailblazer

1, 2

1, 3

What is a transformation?

For each of these shape properties, decide whether they will change or not change when undergoing rotations, translations or reflections:

The position of the shape

The size of the shape

The number of sides of the shape

The angles of the shape

What does a translation do to a shape?

What does a reflection do to a shape?

What does a rotation do to a shape?

Fluency Ex 1

Explorer 1–3

4–8

5–8

Do the pictures show a translation? a

578

5–8

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 2

Ex 3

Do the pictures show a reflection? a

Do the pictures show a rotation? a

14.01 Transformation of shapes mathspace.co

579

Identify whether the transformation is a reflection, rotation or translation: a

Reasoning 9

10, 12

9, 10, 12

Explain which two transformations would be needed to get from flag A to flag B.

9–13 y 7 6 5 4 B

3 A

2 1

x 1

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Mathspace WA Curriculum – Year 7 mathspace.co

Explain why we cannot use a reflection to transform flag A to flag C.

y 7 6

5 A

4 3

2 1

x 1

Riccardo created the tower with building blocks. Is it possible to use a single transformation to place block E and block N at the highest point of the tower? Explain your answer.

D S K F I C H

B L M O E Q A J G N

Which of these figures is a translation of the other? Explain your answer. G

D F

B E

Consider this figure of three carts on a roller coaster track: a

Explain why cart A can move to cart B using translation, but not to cart C.

How would you adjust the track to be able to move cart A to cart C using translation?

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Problem-solving 14

14, 15

14–16

Chantelle wants to animate an object for her slideshow presentation.

She wants the left arm of the person to be reaching for the apple as shown in the figure. What transformation can she apply to achieve this effect?

She also wants to add the effect so that the apple falls to the ground. What transformation can she apply to achieve this?

The moon is in tidal lock to the earth. This means the same part of the moon is facing the earth in every position.

What transformation can be used to get from the moon at position A to the moon at position B?

Moon

Earth

Moon

C Moon B

Katie is lost in a large amusement park and is trying to find her way back to the entrance.

Entrance

Describe the transformation(s) for Katie to move from the roller coaster location to the park entrance.

Pendulum ride Carousel

Dodgems

Ferris wheel Roller coaster

14.02

Translations

After this lesson, you will be able to… • define translation and explain how it affects the position of a shape. • perform and describe translations accurately.

Translations A translation is a type of transformation where the object moves up, down, left, right or a combination of these. We can think of sliding a shape across a page or screen as a translation. 582

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Interactive exploration

Discover this concept in action online

mathspace.co

To describe a translation, we count the number of units it moves left, right, up or down. It can be helpful to choose a specific point on the shape, such as a corner. Then, we can count the units between the original point and the corresponding point on the new shape.

1 unit

In the figure, triangle A has been translated 4 units to the right to produce triangle B.

4 units right

Example 1 Plot the translation of the point by moving it 11 units to the left and 9 units down.

4 3 2 1 −3 −2 −1 −1

x 1 2 3 4 5 6 7 8

−2 −3 −4

Apply the idea 5 4

y 11 units left

9 3 units 2 down 1 −3 −2 −1 −1

4 3 2 x

1 2 3 4 5 6 7 8

−2

1 −3 −2 −1 −1

1 2 3 4 5 6 7 8

−2

−3

−4

Move the point 11 units to the left, then 9 units down.

We end up at the point (-3, -4).

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Example 2 y

Draw the resulting figure B after translating figure A 3 units to the right and 1 unit down.

8 7 6 5 4 3 A

2 1

x 1

Apply the idea y

3 units right

1 unit down

4 3

3 A

2 1

x 1

Move the points of figure A 3 units to the right and 1 unit down.

1 1

x 8

The resulting figure B with the following points: (5, 1), (5, 4), (7, 4) and (7, 1).

Idea summary A translation is a type of transformation where the object moves up, down, left, right or a combination of these. With translations, the size, shape and rotation of the object does not change. To describe a translation, we count the number of units it moves left, right, up or down.

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14.02 Practice questions Understanding 1

Explorer 1–4

Trailblazer

3, 4

Does the given set of images show a translation? a

Adventurer

Which of these statements is true? • When objects are translated, the size of the image remains unchanged. • When objects are translated, the size of the image could change.

Which of these statements is true? • When a point is translated, the values of both the x and y-coordinates will change. • When a point is translated, the values of both the x and y-coordinates might change.

Consider the point M (4, −2) plotted on a Cartesian plane.

If M is translated to each of these points, determine in which direction the point was translated: a

(4, 6)

(1, −2)

(4, −6)

(7, −2)

8 4 2

−8 −6 −4 −2 −2

6 M

−4 −6 −8

Fluency 5

5–7, 10

6–11

7ef, 8–11

On the given Cartesian plane, point A (5, −2) is translated to point A’ (−3, 1). a

How many units to the left has A been translated?

How many units up has A been translated?

A′

5 4 3 2 1

−4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5 6 A

14.02 Translations mathspace.co

585

A point on the Cartesian plane is translated 4 units up.

5 4 3 2 1

Write down the coordinates of its new position.

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Ex 1

x 1

2 3 4 5

Plot the translation of the point by using the given instruction: a

Translate the point 3 units to the right.

Translate the point 4 units to the left.

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

Translate the point 7 units up. 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

Mathspace WA Curriculum – Year 7 mathspace.co

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

2 3 4 5

586

2 3 4 5

Translate the point 4 units down. 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

Translate the point 2 units to the right f and 4 units down.

Translate the point 5 units to the left and 6 units up.

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

x 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

2 3 4 5

A point has been translated 2 units left. The translated point is shown on the Cartesian plane.

x 1

2 3 4 5

Plot the starting position of this point before it was translated.

7 6 5 4 3 2 1

x 1

Point A is translated 3 units down and 4 units to the right, where it sits at point (5, −2). Write down the coordinates of the original point A.

Ex 2

Draw the resulting figure B after translating figure A using the given instruction: a

2 units up

2 units right and 1 unit up

y y 8

3 A

2 1

x 1

14.02 Translations mathspace.co

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4 units up

7 units down

5 4

x 1

5 units left and 3 units down

4 units right and 2 units up

y y 8

7 A

6 5

1 2

4 A

2 3 4 5 6 7 8 9 10

1 unit left and 1 unit down

3 2

2 1

x 1

588

2 units down

x 1

Mathspace WA Curriculum – Year 7 mathspace.co

x 1

A triangle has been translated 3 units left. The translated triangle is shown on the Cartesian plane. Plot the starting position of the triangle before it was translated.

8 7 6 5 4 3 2 1

x 1

Reasoning 12

13ab

12, 13

On a chessboard, a knight must move in an L shape. That is, 1 unit in any direction and then 2 units in a perpendicular direction. If a knight is at square E now, what labelled square could it have been at two moves ago?

Each of these scenarios involves a combination of translations that can be replaced with a single translation. Determine the single translation that would achieve the same final point as the given combination of translations: a

The point (1, 1) is translated 2 units left and 1 unit up, followed by a translation of 5 units right and 2 units down.

The point (6, −2) is translated 3 units right and 3 units up, followed by a translation of 2 units left and 1 unit down.

The point (−1, 4) is translated 4 units right and 6 units down, followed by a translation of 6 units left and 2 units up.

The point (−5, −6) is translated 4 units left and 4 units down, followed by a translation of 10 units right and 11 units up.

14.02 Translations mathspace.co

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Problem-solving 14ab, 15a, 16a 14

14–17

Describe the translation from point A to A’: a

A (2, 4) to A’(2, 7)

A (5, 3) to A’(0, 3)

A (−4, 10) to A’(−1, 6)

A (−2, −2) to A’(3, 6)

Describe the translation of triangle ABC to triangle A’B’C’: a

6 C′

C′ C

B′

2 x

−3 −2 −1

1 2 3 4 5 6 7 8 9

−6 −4 −2

Describe the translation of the trapezium ABCD to the trapezium A’B’C’D’: y

5 4 3 2 1 D′ −4 −3 −2 −1 −1 −2 A −3 −4 −5 −6 D −7 A′

B′

4 2

−8 −6 −4 −2 B′ C′ −2

C′ 3

−4

−6 A

−8 A′

D′−10

Describe the translation from figure B to figure A: a

6 A

3 B x 2

B A

1 1

4 2

590

B′

8 6

A′

Mathspace WA Curriculum – Year 7 mathspace.co

x 1

y 11 10 9 8 7 6 5 4 3 2 1

d 6

4 A

3 2

B x 1

14.03

x 1

Reflections

After this lesson, you will be able to… • identify a reflection of a shape or an object. • perform reflections of shapes across the x or y-axis.

Reflections Reflections are formed by flipping an object or point over a line of reflection. The line of reflection can be horizontal, vertical or diagonal.

Interactive exploration

Discover this concept in action online

mathspace.co

When you see a reflection, it is like looking at a mirror image of something. The reflected image will be the same size as the original object, but it will face the other way. To figure out if an object has been reflected, we can look at how far away the important points of the shapes are from the line of reflection. If the corresponding parts of the shapes are the same distance from the line, then we know a reflection has occurred.

1 unit

2 units

In the figure, triangle A has been flipped over the line of reflection, resulting in triangle B. Notice that the corresponding points on A and B are the same distance from the line of reflection.

14.03 Reflections mathspace.co

591

Example 1 Draw the result of the reflection of this triangle across the y-axis. 10 8 6 4

2 −10

−8

−6

−4

−2

−2 −4 −6 −8 −10

Create a strategy The reflected points must be the same distance from the line of reflection.

Apply the idea The point (−2, −1) is 2 units away from the y−axis, so the reflected point is (2, −1). The point (−3, 9) is 3 units away from the y−axis, so the reflected point is (3, 9). The point (−5, −7) is 5 units away from the y−axis, so the reflected point is (5, −7). 10

3 units 8 3 units 6 4 2 −10

−8

−6

−4

−2 2 2 units −2 2 units −4 −6

5 units

−8 −10

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Mathspace WA Curriculum – Year 7 mathspace.co

5 units

Idea summary A reflection is the result of flipping an object or shape across a line of reflection. When an object is reflected, the reflected image will be the same size as the original object, but the reflected image will face in a different direction to the original object. Every point on the object or shape has a corresponding point on the reflected image and the corresponding points will be the same distance from the line of reflection.

14.03 Practice questions Understanding 1

Explorer

Trailblazer

1–3

3–4

Choose the picture that shows a reflection. A

Adventurer

1–3

Choose the picture that shows a reflection. A

14.03 Reflections mathspace.co

593

Does the image show a reflection? a

Determine whether these diagrams show a reflection across the y-axis: a

x −3

−2

−1

−4 −3 −2 −1

2 1

x −6

594

−4

−2

Mathspace WA Curriculum – Year 7 mathspace.co

−4 −3 −2 −1

Fluency 5

5, 6ad, 7ab, 8–11

5, 6acd, 7a-c, 8–11, 12ac

6–9, 12, 13

Reflect these shapes across the y-axis: a

14.03 Reflections mathspace.co

595

Reflect these shapes across the x-axis: a

Ex 1

Plot the reflection of the given triangle across the y-axis:

y 6 4 2 −5 −4 −3 −2 −1 −2 −4 −6 −8 −10

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Mathspace WA Curriculum – Year 7 mathspace.co

x 1

2 3 4 5

Plot the reflection of the given triangle across the x-axis:

y 4 2 x −8 −6 −4 −2

−2 −4

Plot the reflection of the given triangle across the y-axis:

7 y 6 5 4 3 2 1 −6 −5 −4 −3 −2 −1−1

x 1 2 3 4 5 6

−2 −3 −4 −5 −6 −7

Plot the reflection of the given triangle across the x-axis:

y 6 4 2 x −4 −2

A (8, −6); y-axis

−2 −4 −6

For each coordinates: • Plot the point A on a Cartesian plane. • Plot point A’, a reflection of point A, across the indicated axis. a

A (8, 5); x-axis

A (−10, −6); x-axis c

A (−5, −3); y-axis

Consider the line segment AB, where the endpoints are A (−9, −10) and B (4, 3). a

Plot the line segment AB on a Cartesian plane.

Plot the reflection of the line segment AB across the x-axis. 14.03 Reflections mathspace.co

597

Reasoning 14

14, 15

This diagram shows reflections across line L: a

8 cm

Find the value of: i

iii

bº 9 cm

eº

dº

L 150º

Where will the reflection of point K lie?

8º 37º a cm

c cm

Determine whether each of these statements is true or false: a

When a figure is reflected across the x-axis, only the sign of the y-value changes.

When a figure is reflected across a vertical line, only the y-value changes.

When a figure is reflected, the original figure and image have the same size and shape.

When a figure is reflected, the original figure and image have the same orientation.

A figure is reflected across a line. If the reflected figure is then reflected across the same line again, it will return to the original figure.

Problem-solving 16, 17

16, 17

16–18

Point A has coordinates (−3, 4). Find the coordinates of the image point A’ after a reflection in the y-axis followed by a reflection across the x-axis.

Let A (−2, 1), B (2, 4) and C (4, 2) be the three vertices of a triangle. If this triangle is reflected across the x-axis, what will be the new vertices A’, B’ and C’?

You plan to have a hamburger from point B, while your friend prefers a hotdog from point H. Referring to the top-down view of the map, it shows the building in a green rectangle and the car park in a blue rectangle. Which of the plotted points on the map minimises the walking distance from the car park for both of you? Considering the locations of B and H, and your knowledge of reflection, explain how you chose your answer.

y 6 B

4 2

Mathspace WA Curriculum – Year 7 mathspace.co

−8 −6 −4 −2 2 C −2 O −4 −6

598

H x 4

14.04 Rotations After this lesson, you will be able to… • perform and describe rotations of shapes using a given centre, angle of rotation and direction. • interpret the effects of the rotation on the shape.

Rotations Another type of transformation, known as a rotation, comes from turning an image about a fixed point. This fixed point is known as the centre of rotation. The centre of rotation could be the origin, a point on the image or a point not on the image.

Interactive exploration

Discover this concept in action online

mathspace.co

The sizes of the image and the original object are the same as it turns around the centre of rotation. To perform a rotation we need: • A point for the centre of rotation • The number of degrees to turn • The turning direction (either clockwise or anticlockwise) To rotate an object, we can use a protractor to measure the correct angle of rotation, so we can draw the transformation. For example, to rotate a triangle 90° clockwise about the origin, each vertex of the triangle and its rotated image must create an angle of 90° with the origin. The rotated point will be the same distance from the centre of rotation as the original. y

1 −2

−1 −1

x 1

−2

−1 −1

−2

−3

Rotate the first vertex 90° clockwise

x 1

Rotate the second vertex 90° clockwise

14.04 Rotations mathspace.co

599

1 −2

−1 −1

x 1

−2

−1 −1

−2

−3

Rotate the third vertex 90° clockwise

x 1

Connect the points

Notice how both triangles are the same size, but the rotated triangle is now facing a different direction from the original triangle. Using a protractor, we can confirm that the corresponding vertices of the triangles form a 90° angle with the centre of rotation.

Example 1 Draw the shape after it has been rotated 90° clockwise about the origin.

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Create a strategy Use a protractor to measure the correct angle of rotation and draw the transformation.

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Apply the idea 5 4 3 2 1

A −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Rotate the first vertex 90°clockwise 5 4 3 2 1

A −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Rotate the third vertex 90°clockwise

5 4 3 2 1

A −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Rotate the second vertex 90°clockwise 5 4 3 2 1

A −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x B 1 2 3 4 5

Connect the points.

Reflect and check After a rotation of 90 degrees clockwise: (-2, 4) became (4, 2) (-3, 1) became (1, 3) (-4, 5) became (4, 5) Can you identify the pattern? What would happen if it were rotated anticlockwise instead, what would the pattern be then?

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601

Example 2 Draw the result of a rotation by 180° clockwise about point A. 13 12 11 10 9 8 7 6 5 4 3 2 1

x 1 2 3 4 5 6 7 8 9 10 11 12 13

Create a strategy

Apply the idea

Draw horizontal and vertical lines using A as the centre, then recall that the first quarter of rotation is 90°, second quarter is 180°, third quarter is 270° and last quarter is 360°.

180° is half of the full turn of 360°, so the image should be halfway around a 360° rotation from the original image. Ensuring that the distance from the centre of rotation remains the same. The rotation is: 13 12 11 10 9 8 7 6 5 4 3 2 1

180º A

x 1 2 3 4 5 6 7 8 9 10 11 12 13

Idea summary Rotation of a shape can be done by turning it about the centre of rotation. The corresponding parts of the original object and the rotated image will form the same angle of rotation with the centre of rotation. Rotating an object does not change the size of the shape, but the rotated image will face a different direction to the original shape.

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Did you know?

Rotating gears help clocks keep perfect time. Inside a mechanical clock or watch is a network of tiny interlocking gears, each designed to rotate at a specific angle and speed. These gears don’t just spin randomly—they are carefully engineered to follow strict rules of geometric rotation to measure time accurately. In a clock’s mechanism, each gear rotates around its central axis, much like how a shape rotates about a centre point in geometry. The angle through which each gear rotates depends on both the number of teeth it has and its size. These precise relationships ensure that all the gears work together smoothly to keep accurate time.

14.04 Practice questions Understanding 1

Adventurer 2, 3

Trailblazer 2, 3

Does each picture show a rotation? a

Explorer 1, 2

Complete these statements: a b c d

To rotate a point 90° clockwise is the same as rotating it ⬚ anticlockwise.

To rotate a point 180° anticlockwise is the same as rotating it ⬚ clockwise.

To rotate a point 270° clockwise is the same as rotating it ⬚ anticlockwise. To rotate a point 45° anticlockwise is the same as rotating it ⬚ clockwise.

14.04 Rotations mathspace.co

603

Consider the △ABC: y B

6 A

x −9

−6

−3

−3 −6 −9

Choose the correct representation of △A’B’C’, the result of rotating △ABC 180° clockwise about the origin. A

3 x

−9

−6

B′

x −9

−6

−3

A′−3

−6 C′

−3 A′ −3

−6

C′

−9

B′

3 x

−9

−6 −3 B′ −3

C′ −6 A′

604

−9

Mathspace WA Curriculum – Year 7 mathspace.co

x −9

−6

−3

3 −3

A′

B′

−6 C′ −9

Fluency 4

4, 5, 8

4–8

Find the coordinates of point B when it is rotated 90° clockwise about the origin. 5 4 3 2 1

y B

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Ex 1

4–8

2 3 4 5

Draw the shape after it has been rotated 180° anticlockwise about the origin. 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

14.04 Rotations mathspace.co

605

Suppose all four triangles are rotated 270° clockwise about the origin.

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 R −5

y P x 1 2 3 4 5 S

Which image would correspond to the result of this transformation? A

y S

−5 −4 −3 −2 −1 −1 −2 −3 P −4 −5

P R x

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 P −5

D R R x

1 2 3 4 5

Mathspace WA Curriculum – Year 7 mathspace.co

5 4 3 2 1

y S x

−5 −4 −3 −2 −1 −1 −2 −3 −4 Q −5

1 2 3 4 5

y S

606

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 S −5

1 2 3 4 5 R

y Q x 1 2 3 4 5 P

Which shape corresponds to a rotation of triangle R by 270° clockwise about the origin?

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5 T Ex 2

Draw the flag after it has been rotated clockwise: a

90° about point A

180° about point A

270° about point A

1 2 3 4 5 R

y 5 4 3 A

2 1

x 1

Reasoning

10, 11

9–11

If a figure is rotated by an angle of less than 360°, can it still look like the same figure with the same orientation and position? Justify your answer by drawing a diagram.

Aneta has attempted to rotate the segment AB 90° anticlockwise about the origin. a

Has Aneta made an error? Explain how you know.

Find the coordinates of A’ and B’ if the segment AB is rotated 90° anticlockwise about the origin.

3 2 B

1 −4 −3 −2 −1

−1

−2

x 3

B′

−3

A′

−4

Akilah was looking at the faces on a die. She noticed that the face with a single dot looks the same when rotated by 180°. a

Akilah hypothesised that each face looks the same when rotated 180°. Is she correct? Explain your reasoning.

Akilah also hypothesised that each face looks the same when rotated 90° in either direction. Is she correct? Explain your reasoning.

14.04 Rotations mathspace.co

607

Problem-solving 15, 16 12

13–16

12–16

Draw the shape after it has been rotated 180° clockwise about point A.

13 12 11 10 9 8 7 6 5 4 3 2 1

x 1 2 3 4 5 6 7 8 9 10 11 12 13

For each of these where shape A has been rotated to the position of shape B around point O: i

Find the clockwise angle of rotation.

Find the anticlockwise angle of rotation.

9 8

6 5 O

x 1

Consider this image: a

Which segment represents a clockwise rotation of segment PQ by 90° about point O?

Which segment represents an anticlockwise rotation of segment XY by 270° about point O?

x 1

y 11 10 9 8 7 P Y 6 5 4 3 2 1

J Q

K U

V X

1 2 3 4 5 6 7 8 9 10 11

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Mathspace WA Curriculum – Year 7 mathspace.co

The centre of rotation for these figures is point A. Identify how many degrees and in which direction each figure has been rotated: a

A A

Avery is on the corner of Argyle Street and Campbell Street. Argyle Street runs north-south and Campbell Street runs east-west. He has a map with a compass at the top depicting that the top of the map is north. If Avery is facing south on Argyle Street, how many degrees and in what direction does he need to turn the map so the top of the map is aligned with the direction he is facing? Explain how you know.

14.04 Rotations mathspace.co

609

14.05 Multiple transformations After this lesson, you will be able to… • analyse how the order in which transformations are applied can affect the final result. • apply multiple transformations to shapes and objects to create interesting patterns and results. • reflect, rotate and translate shapes on the Cartesian plane using multiple transformations.

Multiple transformations A shape or object may be transformed more than once. A list of transformations that are performed one after the other is called a composition of transformations. The order in which we apply the transformations can greatly impact the final result.

Exploration Use this triangle on the Cartesian plane to answer the questions: 4

1. Reflect the triangle across the x-axis, then rotate it 90° clockwise about the origin. What are the coordinates of the vertices? 2. Rotate the triangle 90° clockwise about the origin, then reflect it across the x-axis. What are the coordinates of the vertices?

3 2 1 −4 −3 −2 −1

−1

3. How does the order of transformations affect the final result?

−2

4. Beginning again with the original triangle, translate it left 5 units, then reflect it across the x-axis. What are the coordinates of the vertices?

−4

x 1

−3

5. Reflect the triangle across the x-axis first, then translate it left 5 units. What are the coordinates of the vertices? 6. Did the order of transformations affect the final result from steps 4 and 5?

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The order in which the transformations are applied is important because the transformed object could end up in a different position if the transformations are applied in a different order. Let’s take a look at this example.

Start with rectangle ABCD, perform a translation of 5 units to the left and 11 units up, followed by a reflection in the y-axis.

After the translation, we have rectangle A'B'C'D'.

y A′

B′

B′′

A′′

D′

C′

C′′

D′′

After the reflection, we have rectangle A"B"C"D". The number of ticks on each vertex of the shape allows us to keep track of the number and order of transformations.

y B′′

A′′

C′′

D′′

B′

A′

C′

D′

Notice that if we performed the reflection and then the translation, the final rectangle would end up in a different position.

14.05 Multiple transformations mathspace.co

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Example 1 Plot the new point on the plane that results from translating the given point 5 units downward and then rotating it 90° anticlockwise about the origin. 6

5 4 3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Create a strategy Perform the translation, then the rotation. Remember that when a point is rotated 90° anticlockwise about the origin, the angle of rotation between the original point and the rotated point is 90° and the centre of the angle is at the origin.

Apply the idea 6

(3, 6)

2 (3, 1)

1 −1

−2

−4 −3 −2 −1

−1

−3

−4

Mathspace WA Curriculum – Year 7 mathspace.co

x 1

−2

−3

Performing the translation first, we slide the point downwards by 5 units. This means that only the y-coordinate will change.

612

(3, 6)

(−1, 3)

2 −4 −3 −2 −1

Since the coordinates of the point after translation are (3, 1), the coordinates of the point after rotating 90° anticlockwise are (−1, 3). The coordinates of the new point will be (−1, 3).

Example 2 Point A (−9, 7) is rotated 90° clockwise about the origin to form point B. a What are the coordinates of point B?

Create a strategy

Apply the idea

Remember that when a point is rotated 90° clockwise about the origin, the angle of rotation between the original point and the rotated point is 90° and the centre of the angle is at the origin.

When a point with coordinates (−9, 7) is rotated 90° clockwise, its coordinates become (7, 9). The coordinates of point B are (7, 9).

b Point B is then translated to the left so that it lies on the same vertical line as point A. How many units to the left is point B translated?

Create a strategy Use the fact that in order for the two points to lie on the same vertical line, they must have the same x-coordinate.

Apply the idea The x-coordinate of point A is −9 and the x-coordinate of point B is 7. Point B is then translated to the left so that it lies on the same vertical line as point A. To determine the units to the left of translated point B, we do this by finding the distance between 7 and −9. The distance between 7 and −9 is 7 + 9 = 16 units. The point B has been translated 16 units to the left.

Example 3 Plot the given shape that would result when it is rotated 180° clockwise about the origin and then reflected across the y-axis.

4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

14.05 Multiple transformations mathspace.co

613

Create a strategy Perform the rotation first. Recall that the first quarter of rotation is 90°, the second quarter is 180°, the third quarter is 270° and the last quarter is 360°. Then, reflect the points from the rotated shape across the y−axis. Recall that the reflected points must be the same distance from the line of reflection.

Apply the idea 5

4 3 2 1

180º x

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

2 3 4 5

The rotated shape with points

(1, −1), (5, −1) and (3, −5) is now to be reflected across the y−axis.

3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

180° is half of the full turn of 360°, so the image should be halfway around a 360° rotation from the original image.

2 3 4 5

The point (1, −1) is 1 unit away from y−axis, so the reflected point is (−1, −1). The point (5, −1) is 5 units away from y−axis, so the reflected point is (−5, −1). The point (3, −5) is 3 units away from y−axis, so the reflected point is (−3, −5).

4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

614

x 1

2 3 4 5

Mathspace WA Curriculum – Year 7 mathspace.co

Here is the shape after the rotation of 180° clockwise about the origin and reflection across the y-axis.

Idea summary A shape or object may be transformed more than once. When we apply multiple transformations to a shape, we perform them one after another. The order that you perform the transformations in can affect the result.

14.05 Practice questions Understanding

Adventurer 2

Trailblazer 2

Name and explain the three main types of transformations.

If the same transformations are applied to two shapes, but in different orders, will they end up with the same image? Why or why not?

Fluency Ex 1

Explorer

3–10, 12–14, 16–17

3–9, 13, 14, 16, 17, 19, 20

Plot the point on the plane that results from translating the given point 2 units downward and then rotating it 90° anticlockwise about the origin.

3–9, 11, 12, 15–20

4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Point A on the plane is to undergo two successive transformations:

Plot point A’, which is the result when point A is translated 4 units left.

Plot point A”, which is the result from reflecting point A’ across the y-axis.

2 3 4 5

4 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

14.05 Multiple transformations mathspace.co

615

Point A on the plane is to undergo two successive transformations: a b

Plot point A’, which is the result when point A is rotated 180° anticlockwise about the origin. Plot point A”, which is the result from reflecting point A’ across the x-axis.

4 3 2

1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Point A on the plane is to undergo two successive transformations:

Plot point A’, which is the result from reflecting the point A across the x-axis.

Plot point A”, which is the result when point A’ is rotated 90° clockwise about the origin.

Point A is marked on the Cartesian plane: a

If point A is rotated 270° clockwise about the origin, what are the coordinates of point A’?

Plot the point that would result when A’ is translated 6 units down to make point A”.

2 3 4 5

y A

4 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

2 3 4 5

3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

For the point on the plane (−6, 9), write down the coordinates of the point that will result from reflecting the point across the x-axis followed by a rotation of 180° clockwise about the origin.

For the point on the plane (−2, −1) write down the coordinates of the point that will result from a rotation of 90° anticlockwise about the origin followed by reflecting the point across the x-axis.

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Mathspace WA Curriculum – Year 7 mathspace.co

Write down the coordinates of the point that will result from a translation of 2 units right followed by a rotation of 90° anticlockwise about the origin for these points: a

Ex 2

(4, 5)

(−7, −6)

For the point on the plane (1, 4), write down the coordinates of the point that will result from a rotation of 270° clockwise about the origin followed by a translation of 4 units left and 3 units down.

For the point on the plane (3, −6), write down the coordinates of the point that will result from a rotation of 180° anticlockwise about the origin followed by reflecting the point across the y-axis.

For the point on the plane A (5, −9), write down the coordinates of A”, that result from reflecting the point across the x-axis and translating it 5 units down.

For the point on the plane A (2, −8), write down the coordinates of A”, that result from reflecting the point across the y-axis and translating it 4 units left.

Point A (−10, 6) is rotated 180° clockwise about the origin to form point B.

Write down the coordinates of point B.

Point B is then translated to the left so that it lies on the same vertical line as point A. How many units to the left has point B been translated?

Plot the given shape that will result from reflecting it across the x-axis followed by a translation of 6 units right.

4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Plot the given shape that will result from a translation of 6 units left followed by a rotation of 180° clockwise about the origin.

2 3 4 5

4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

14.05 Multiple transformations mathspace.co

617

Plot the given shape that will result from reflecting it across the y-axis followed by a rotation of 270° clockwise about the origin.

4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Plot the given shape that will result from a rotation of 180° anticlockwise about the origin followed by a translation of 6 units left and 1 unit down.

2 3 4 5

4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Ex 3

Plot the given shape that would result when it is rotated 90° anticlockwise about the origin and then reflected across the x-axis.

2 3 4 5

4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

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Mathspace WA Curriculum – Year 7 mathspace.co

x 1

2 3 4 5

Reasoning 21

22, 23

Plot the point A’, resulting from the combination of both translations on point A.

Describe a single translation that would achieve the same result.

Translation of 4 units left and 2 units up, followed by a translation of 3 units right and 3 units down.

Translation of 3 units left and 4 units down, followed by a translation of 2 units left and 3 units up.

21–23

Point A is marked on the Cartesian plane. For each of these transformations:

Translation of 3 units right and 2 units down, followed by a translation of 1 unit right and 4 units down. Translation of 8 units right and 3 units down, followed by a translation of 7 units left and 6 units up.

4 3 2

1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 2 3 4 5

Here is the flag for Trinidad and Tobago: a

Describe how to map section a1 of the flag onto section a2 using a single transformation.

Describe how to map section a1 of the flag onto section a2 using two consecutive transformations.

Here is a pattern from a piece of Kente fabric, woven in Ghana. It is made up of two shapes; a square each separated by a long thin rectangle.

Describe a sequence of transformations that could be used to recreate the pattern.

14.05 Multiple transformations mathspace.co

619

Problem-solving 25, 26 24

27, 28

Determine two types of transformations that could be used to get from Flag A to Flag B: a

5 B

x 1

24, 26–28

x 1

Determine two types of transformations that would be y needed to get from: 7 a Flag A to Flag B b

Flag A to Flag C

5 A

4 3

2 1

x 1

Identify the new shape that results from rotating the original shape 90° anticlockwise about point O and then translating it 3 units upward.

11 10 9 8 7 6 5 4 3 2 1

Original O 1 2 3 4 5 6 7 8 9 10 11

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Mathspace WA Curriculum – Year 7 mathspace.co

Triangle M is to be rotated 90° clockwise about point O, then translated 3 units left. Which triangle corresponds to the result of this transformation?

M R

Identify the correct square that results from rotating the original square 90° anticlockwise about point Q, reflecting it across the line PR and then translating it 4 units to the left

D C P

E Q

B A Original

Did you know?

Architects use transformations of shapes to design complex buildings! By rotating, scaling and translating shapes, they can create intricate patterns and structures, transforming simple blueprints into iconic skyscrapers and innovative designs. 14.05 Multiple transformations mathspace.co

621

Investigation: Tessellations

Investigate online

mathspace.co

14.06 Line and rotational symmetry After this lesson, you will be able to… • identify the two main types of symmetry: line symmetry and rotational symmetry. • identify lines of symmetry in shapes. • determine the order of rotational symmetry of shapes.

Line symmetry A shape has symmetry if it looks the same before and after a transformation. Symmetries are very common in nature as well as many areas of mathematics. The two main types of symmetry we will explore are line symmetry (or reflection symmetry) and rotational symmetry. A line that reflects a shape onto itself is called a line of symmetry (or axis of symmetry).

Interactive exploration

Discover this concept in action online

A square has four lines of symmetry, all passing through its centre.

1 2

622

mathspace.co

Mathspace WA Curriculum – Year 7 mathspace.co

One way to picture lines of symmetry is with reflections. Another way is to picture folding the shape along a line. Whether reflecting or folding, the resulting shape should perfectly overlap the original.

A shape that has no lines of symmetry is called asymmetric. Neither of these shapes have a line of symmetry.

Example 1 How many lines of symmetry does this figure have?

Create a strategy Draw a line through the shape that makes one half of the shape a mirror reflection of the other half.

Apply the idea For this shape, one such line of symmetry looks like this vertical line. But we can also draw a horizontal line of symmetry to divide the shape into halves. So, the figure has 2 lines of symmetry.

Idea summary A line that reflects a shape onto itself is called a line of symmetry (or axis of symmetry). Asymmetric shapes are shapes without lines of symmetry.

14.06 Line and rotational symmetry mathspace.co

623

Rotational symmetry If a rotated shape perfectly overlaps the original shape after a rotation that is less than 360°, then the original shape has rotational symmetry. The point about which this rotation happens is called the centre of rotation.

Interactive exploration

Discover this concept in action online

mathspace.co

Shapes with rotational symmetry match their original position multiple times during a full 360° turn. The number of times a shape matches its original position during a full turn is called the order of rotational symmetry. For example, a square has rotational symmetry because when you turn it by 90°, it looks the same as before. You can do this four times in a full 360° turn, so a square has a rotational symmetry order of 4. A right triangle only looks the same after turning a full 360°, this is referred to as having an order of rotational symmetry of 1. Rotational symmetry exists only where shapes have an order of rotational symmetry of 2 or higher.

Example 2 Which of these shapes have rotational symmetry? If it does, what is the order of rotational symmetry? A

Create a strategy If a rotated shape perfectly overlaps the original shape after a rotation that is less than 360°, then the original shape has rotational symmetry.

Apply the idea All the shapes have rotational symmetry except option B. The answers are options A, C and D.

Shape A has an order of rotational symmetry of 2.

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Mathspace WA Curriculum – Year 7 mathspace.co

Shape C has an order of rotational symmetry of 4.

Shape D has an order of rotational symmetry of 6.

Example 3 For each shape, determine if it has line symmetry, rotational symmetry, both types or no symmetry at all. a

Apply the idea The shape can be divided into two equal parts by a vertical line, but it does not have rotational symmetry. This shape has line symmetry.

Apply the idea This is an irregular shape. It cannot be divided into two equal parts by a line and does not have rotational symmetry. This shape has no symmetry.

Apply the idea The shape can be divided into two equal halves with a vertical line, a horizontal line and diagonal lines. It has rotational symmetry of order 4. This shape has both line and rotational symmetry.

14.06 Line and rotational symmetry mathspace.co

625

Idea summary A shape has rotational symmetry if the rotated shape perfectly overlaps the original shape after a rotation that is less than 360°. The point about which this rotation happens is called the centre of rotation. The number of times a shape can rotate onto itself in a full 360° rotation is the order of rotational symmetry.

14.06 Practice questions Understanding 1

Adventurer 1–3

Trailblazer 1–3

Determine which statements describe rotational symmetry and which describe line symmetry. a

Creates a reflection of one half of the shape to the other

Is similar to a flip transformation

Is similar to a turn transformation

Has an order, that indicates how many times the same shape is represented in a 360° turn.

One shape can have many of them

Do all shapes have line symmetry? Give two examples to explain why or why not.

Do all shapes have rotational symmetry? Give two examples to explain why or why not.

Fluency Ex 1

Explorer 1–3

626

4–10, 13–15

4–8, 11, 12, 14, 15

4–7, 11, 12, 15, 16

How many lines of symmetry do these shapes have? a

Which of these single-digit numbers have line symmetry?

Mathspace WA Curriculum – Year 7 mathspace.co

Determine whether these letters have rotational symmetry: a

Ex 2

Determine whether these diagrams have rotational symmetry. a

Do these road signs have rotational symmetry? a

Ex 3

Identify whether these diagrams have line symmetry, rotational symmetry, both or neither: a

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Complete each picture in these diagrams so that the dotted line is a line of symmetry. a

The points B, D, F and H represent the midpoints of the sides of the rhombus. Write down the pairs of points that can be joined to form a line of symmetry for the rhombus.

The points F, G, H and E represent the midpoints of the sides of the rectangle. Write down the pairs of points that can be joined to form a line of symmetry for the rectangle.

Sketch the lines of reflectional symmetry on each shape: a

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Consider the road sign: a

What is the order of rotational symmetry?

Does the road sign have line symmetry?

Mathspace WA Curriculum – Year 7 mathspace.co

F E

What is the order of rotational symmetry for each of these shapes? a

If a shape has a rotational symmetry of the given order of symmetry, what is the size of each angle of rotation? a

order 2

Reasoning

18, 20

order 4

17, 19, 20

Do all isosceles triangles have rotational symmetry? Explain your reasoning.

Do all equilateral triangles have rotational symmetry? Explain your reasoning.

Do all regular polygons have rotational symmetry? Explain your reasoning and provide an equation for the angle of rotation required to get a symmetrical shape.

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Do either of these shapes have rotational symmetry? Explain your reasoning using coordinates on the Cartesian plane. a

4 C

2 A

1 −1

x 1

−4 −3 −2 −1

−2 A

−3

−4

Problem-solving 21, 23 21

−4 −3 −2 −1

x 1

−1

−2 −3 −4

21–23

22–24

Consider all the letters of the alphabet, both uppercase and lowercase. a

List all the uppercase letters that possess a line of symmetry.

Determine if any of the letters have more than one line of symmetry and if yes, which letters have multiple lines of symmetry.

The shape is constructed from five identical rhombuses: a

How many lines of symmetry does this shape have?

If each rhombus has a side length of 10 cm, find the perimeter P of the entire shape.

The toy fan spins in an anticlockwise direction. During one complete spin, determine the size of the angles of rotation where the fan has rotational symmetry.

Dave wants to get a heavy rectangular box from point A to point B by repeatedly turning it over. The width of the box is 30 cm and its length is twice its width.

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The diagram shows the distance travelled after the first and second turn. After 3 turns, what distance along the ground has been covered starting from point A?

If the distance from point A to point B is 1620 cm, how many times must the box be turned?

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14 Chapter review 1

Identify whether these can change when rotating, translating or reflecting a shape: a

The position of the shape

The size of the shape

The number of sides of the shape

The angles of the shape

Which of these statements is true? • When a point is translated, the values of both the x and y-coordinates might remain the same. • When a point is translated, the values of both the x and y-coordinates might change.

The point H (2, −2) plotted on a Cartesian plane. 8

If H is translated to each point, describe the vertical or horizontal movements of the point: a

(−1, −2)

(6, −4)

(3, 5)

(8, −2)

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

Consider the points plotted on the Cartesian plane: a b c d

What are the coordinates of point A if it is translated 2 units up?

3 B

What are the coordinates of point B if it is translated 2 units left?

2 1

What are the coordinates of point C if it is reflected over x-axis?

−4 −3 −2 −1

What are the coordinates of point D if it is reflected over the y-axis?

x 1

−1

A point has been translated 4 units left. The translated point is plotted on the Cartesian plane. Plot the starting position of this point before it was translated.

−2

−3 −4

8 7 6 5 4 3 2 1

x 1

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Write down the original coordinates of these points after the translation: a

Point A is translated 3 units down and 4 units to the right, where it sits at point B (5, −2).

Point X is translated 4 units up and 5 units to the left, where it overlaps point Y (−10, 5).

Point M is translated 2 units up and 4 units to the right, where it overlaps point N (5, 6).

A triangle has been translated 8 units left. The translated triangle is plotted on the Cartesian plane.

7 6 5 4 3 2 1

Plot the starting position of the triangle before it was translated.

−7−6−5−4−3−2 −1 −1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Each of the scenarios involves a combination of transformations that can be replaced with a single transformation. Find the single transformation that would achieve the same final point as the given combination of transformations: a

The point (3, 4) is translated 4 units left and 3 unit up, followed by a translation of 7 units right and 5 units down.

The point (3, 2) is reflected across the y-axis, followed a reflection across the x-axis.

Plot the reflection of the given triangle across the y-axis:

6 4 2

−5 −4 −3 −2 −1 −2

1 2 3 4 5

−4 −6 −8 −10

Plot the reflection of the given triangle across the x-axis:

y 8 6 4 2 −8 −6 −4 −2 −2 −4 −6

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x 2 4 6 8 10

Consider the point A (8, 5). a

Plot point A on a Cartesian plane.

If point A is reflected across the y-axis to produce point B, what will be the coordinates of point B?

Complete the statements: a b c d

To rotate a point 75° clockwise is the same as rotating it ⬚ anticlockwise.

To rotate a point 180° anticlockwise is the same as rotating it ⬚ clockwise.

To rotate a point 105° clockwise is the same as rotating it ⬚ anticlockwise. To rotate a point 60° anticlockwise is the same as rotating it ⬚ clockwise.

Determine the coordinates of point B when it is rotated 180° clockwise about the origin.

3 2 B

1 −4 −3 −2 −1

−1

C x 3 4

−2 −3 −4

Consider the flag on the Cartesian plane. Draw the flag after it has been rotated clockwise: a

90° about point P

180° about point P

y 5 4 3 P

2 1

x 1

Consider the shape plotted on the Cartesian plane: Draw the shape after it has been rotated by 180° clockwise about point A.

13 12 11 10 9 8 7 6 5 4 3 2 1

x 1 2 3 4 5 6 7 8 9 10 11 12 13

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What is the counterclockwise angle of rotation of shape A to shape B about point O?

y A

8 7 6

5 O

4 3 2 1

x 1

Point A on the plane is to undergo two successive transformations: a

Plot point A′, which is the result when point A is translated 4 units left.

Plot point A″, which is the result when point A′ is rotated 90° clockwise about the origin.

Plot the point on the plane that results from translating the given point 5 units left and then rotating it 90° clockwise about the origin.

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

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−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

2 3 4 5

x 1

2 3 4 5

Plot the triangle that will result when the given triangle is reflected across the y-axis and translated 3 units left and 2 units down.

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

For the point on the plane (−6, 9), write down the coordinates of the point that will result from a translation of 4 units left followed by a rotation of 90° clockwise about the origin.

For the point on the plane (1, 4), write down the coordinates of the point that will result from a translation of 4 units left and 3 units down, followed by a rotation of 90° clockwise about the origin.

For the point on the plane (3, −6), write down the coordinates of the point that will result from a translation of 4 units right and 5 units down, followed by a rotation of 90° anticlockwise about the origin.

Determine whether these shapes have rotational symmetry:

Describe the angle of rotation for rotational symmetries of orders 2, 3, 4 and 5. How can you generalise this for any order of rotation?

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Big ideas The probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.

Curriculum Outcomes • Construct sample spaces for single-stage chance experiments, assign probabilities to the outcomes and predict frequencies for different numbers of trials • Represent primary categorical and numerical data in a Venn diagram, calculate related relative frequencies and interpret results • Analyse, determine and interpret problems in real-world situations that involve assigning a probability to single-stage chance experiments or simulations, statistical measures, stem and leaf plots, dot plots, column graphs and/or Venn diagrams • Conduct repeated single-stage chance experiments and simulations to produce datasets, including through the use of digital tools, for an increasingly large number of trials. Discuss and describe variation and estimated probabilities for outcomes and compare to predictions and theoretical probability, where appropriate

15 Probability Chapter outline 15.01 Language of probability 15.02 Probability as a number 15.03 Theoretical probability 15.04 Relative frequency 15.05 Venn Diagrams Modelling investigation: Recycling probability Investigation: Simulations Chapter 15 review

638 646 652 658 666 678 680

In a standard deck of 52 cards, each card has a 1 in 52 chance of being drawn first!

15.01 Language of probability After this lesson, you will be able to… • group outcomes into events and sort them into categories of likelihood. • apply concepts to real-world scenarios, such as spinning a spinner or rolling a die.

The language of probability Probability is the study of chance and prediction. To make sure our predictions are valid, we need to learn and use the right mathematical language. An experiment is any activity that has a possible result. Each result of an experiment is called an outcome. Events or trials can either be a single outcome or a group of outcomes from an experiment. The list of all possible outcomes of a trial is called the sample space. Experiment Flipping a coin

Sample Space heads, tails

Outcome Getting tails

Rolling a die

1,2,3,4,5,6

Rolling a 6

Event Getting tails (single outcome) Rolling an even number (group of outcomes)

Example 1 If I spin this spinner, what is the sample space?

Create a strategy

Apply the idea

The sample space is the list of all possible outcomes.

When we look at the spinner, we can see that the sample space includes Pig, Ball and Apple.

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Idea summary The language of probability: Trial - a single experiment with different possible results. Outcome - the possible results of a trial. Sample space - a list of all the possible outcomes of a trial.

Describe likelihood We can group outcomes into events. For example, an event when rolling a standard die could be “rolling an even number” or “rolling more than 3”. Each outcome on its own is always an event, but sometimes events do not correspond to any outcomes. We can think about different kinds of events that we care about and sort them into categories of likelihood. The likelihood of an event depends on what happens when you repeat the trial many times. • Impossible: no outcomes correspond to the event • Unlikely: the event happens less than half the time • Even chance: the event happens exactly half the time • Likely: the event happens more than half the time • Certain: every outcome corresponds to the event Here are some examples when rolling a die: Likehood

Event

Impossible

Rolling a 9

Unlikely

Rolling a 1

Even chance

Rolling 4 or more

Likely

Rolling 2 or more

Certain

Rolling between 1 and 6

There are many more possible events we could describe and fitting them into the right likelihood category can take some practice. We will investigate ways we can be precise in the next lesson.

Example 2 A six-sided die is rolled in a trial. What are the chances that the outcome is 3 or more? A Impossible

Unlikely

D Likely

Certain

Even chance

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Create a strategy

Apply the idea

Count the number of outcomes that are 3 or more.

There are 4 outcomes that are 3 or more, which is most of the faces on the die. The chances of that outcome is likely. The correct answer is option D.

Example 3 Which symbol is most likely to land on when the spinner is spun? A Ball

Star

C Apple

Pig

Create a strategy

Apply the idea

Count the symbol that appears most on the spinner.

The spinner has 8 sectors. The ball symbol appears 3 times. Therefore, the symbol that is most likely to land on is the ball. The correct answer is option A.

Idea summary Event - a grouping of outcomes. Each possible outcome is always an event on its own. Likelihood - an event can be: • • • • •

impossible (can never happen) unlikely (happens less than half the time) even chance (happens half the time) likely (happens more than half the time) certain (always happens)

15.01 Practice questions Understanding 1

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Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

Is rolling a die an example of an outcome or a trial?

Is getting a tail when flipping a coin an example of an outcome or a trial?

Mathspace WA Curriculum – Year 7 mathspace.co

When you flip a coin the possible outcomes are Head or Tail. What probability terminology is used for the set of all possible outcomes?

Which of these is true? a

When rolling a die, rolling an even number is an example of an event.

If the event happens more than half the time, we say it is likely.

If no outcomes correspond to the event, we say it is unlikely.

If an event happens exactly half the time, we say it has an even chance.

Fluency 4

5, 6, 8, 9, 11–14, 16, 17

5–7, 9, 11, 12, 16, 17

List the sample space: a

One spin of this spinner

Rolling a standard six-sided die

Vincent rolls a twelve-sided die and writes down whether he rolls 6 or more. If he does, he writes “yes”, otherwise he writes “no”. List the sample space that corresponds to the event where he writes “yes”.

Ex 1

4, 5, 7–17

8 6

Determine whether these events are impossible, unlikely, even chance, likely or certain: a

Winning the lottery

Scoring 140% on an exam

Flipping tails with a coin

Next week will have 8 days

A number greater than 4 is rolled with a standard die.

When a standard die is rolled, we get a number from 1 to 6.

In a prize draw, 8 red tickets and 8 blue tickets are handed out. Choose the best description of the chance of each event: • Impossible • Unlikely

• Even Chance

The winning ticket being black

The winning ticket being red

The winning ticket being red or blue

• Likely • Certain

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Ex 2

Ex 3

Consider the spinner and describe the likelihood of these outcomes: a

A standard six-sided die is rolled. a

Describe the likelihood that the outcome is a 4 or higher.

Describe the likelihood that the outcome is a 7 or higher.

Consider the spinner: a

Are the events of spinning an apple and a star equally likely?

Are the events of spinning a ball and an apple equally likely?

Are the events of spinning a ball and a star equally likely?

Consider the spinner: a

What is the least likely symbol to spin?

Describe the likelihood of spinning a ball.

Which of these pairs of events have an even chance?

642

Spinning an apple or a pig

Spinning a star or a ball

Spinning a pig or a star

Spinning a ball or a pig

Which of these events are equally likely? A

Spinning an apple or a ball

Spinning a ball or a star

Spinning an apple or a pig

Spinning a star or a pig

Mathspace WA Curriculum – Year 7 mathspace.co

Consider the spinner.

Determine whether each event has an even chance:

Spinning an even number

Spinning 7 or more

Spinning 12 or less

Spinning 7 or less

7 11

There are 20 marbles in a jar: 7 red, 10 green and 3 blue. If a marble is chosen at random: a

What colour marble is most likely to be chosen?

What colour marble is least likely to be chosen?

One marble is drawn out of this jar containing 3 black, 3 red and 5 blue marbles. a

Identify the colour of the marble that you are most likely to draw.

Describe the likelihood that you draw a marble that is not Blue.

These three cards are shuffled. One card is then drawn at random:

Describe the likelihood of drawing a Heart.

Describe the likelihood of drawing a number card.

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These four cards are shuffled. One card is then drawn at random:

Describe the likelihood of drawing a Jack, Queen or King.

Describe the likelihood of drawing a card with a number less than 4.

Yuri plays a game using this spinner: a

Describe the likelihood that he spins a number less than 15.

Describe the likelihood that he spins an even number.

Describe the likelihood that he spins the number 4.

Describe the likelihood that he spins the same number three times in a row.

Describe the likelihood that he spins a number between 5 and 14.

Reasoning

19, 22

18–20, 22

12 7

19–22

A twelve-sided die is rolled. Determine the likelihood that the outcome is 8 or higher. Explain your reasoning.

A standard six-sided die is rolled. Which outcome is least likely? Justify your answer.

Rolling a prime number

Rolling a number less than 3

Rolling an even number

Consider drawing a card from a standard deck of cards. Order the events from least likely to most likely. Justify your ordering. 1.

Drawing a black card

3. Drawing a diamond

2. Drawing a seven 4. Drawing a card numbered 2–10

Think of an example where multiple events are equally likely to happen, but the events do not have an even chance of occurring. Use your example to explain the difference between even chance and equally likely.

Marie is watching a car race where there are 48 Holdens and 16 Fords. Each car is equally likely to win. Marie says that it’s impossible for a Ford to win. Is Marie correct? Explain your reasoning.

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Problem-solving 23, 24 23

24, 25ab

23–25

James has a bag of eleven counters numbered 1 to 11. Mae has a bag of twenty counters numbered 1 to 20. Each chooses a counter from their own bag without looking. Determine whether these statements are true or false: a

James is more likely than Mae to choose a counter numbered 11.

They are equally likely to choose a number less than 3.

James is more likely than Mae to choose an odd number.

Mae is less likely than James to choose a counter numbered 16.

Kenneth has a six-sided die. Lisa has a twelve-sided die. They decide to roll both their dice and the winner is the person who rolls the highest. Who is more likely to win? 3 11

Rochelle draws a card from a standard 52 card deck and writes down the suit of the card. a

How many different outcomes could there be?

Are the events equally likely? Explain your reasoning.

Identify different outcomes that are also equally likely.

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15.02 Probability as a number After this lesson, you will be able to… • calculate the probability of an event using the formula. • apply probability calculations to real-world situations. • understand the significance of probability on a number line and how to interpret its value.

Probability as a number The likelihood of an event can be placed on a spectrum from 0 to 1 using fractions or decimals, or from 0% to 100% using percentages: Impossible

Even chance

Certain

Unlikely

0 0%

Likely

0.4 8%

0.5 50%

0.63 72%

100%

A probability can never be less than 0 or more than 1. The larger the number, the more likely it is and the smaller the number, the less likely it is. To find the probability of an event, we can use the formula:

A favourable outcome is the outcomes we want or are looking for. If every outcome is favourable, then we have a probability of 1. If there are no favourable outcomes, then probability is 0.

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Example 1 A probability of

means the event is:

A Impossible

Unlikely

Likely

Certain

Create a strategy Compare the value to 0 and 1.

Apply the idea Probabilities range from 0 to 1. Events with probabilities close to 0 are unlikely and events close to 1 are likely. A probability of

is close to 1 so the event is likely. The correct answer is option C.

Example 2 Which outcomes will have a probability of 25% on this spinner:

Create a strategy Write the percent into a fraction.

Apply the idea The 25% is equal to . It means that the sector will take up one quarter of the spinner. We can see that the apple and pig sectors both take up one quarter of the spinner.

Example 3 A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red. a If the probability of picking a red marble is

, how many red marbles are there in the jar?

Apply the idea We know that the probability of picking a red marble is So there are 4 red marbles.

. That means 4 out of 10 marbles are red.

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b What is the probability of picking a blue marble?

Create a strategy Find the number of blue marble and write the probability as a fraction.

Apply the idea There are 10 marbles in total. We know that 4 of them are red and the rest are blue, so there are 10 − 4 = 6 blue marbles. That means the probability is of picking a blue marble is

Idea summary

If every outcome is favourable, then we have a probability of 1. If there are no favourable outcomes, then probability is 0.

15.02 Practice questions Understanding

Adventurer 1–2

Trailblazer 1–2

Consider a trial where there are six different outcomes and each outcome is equally likely. What will be the probability of one particular outcome?

What is the formula to calculate the probability of an outcome?

Fluency Ex 1

Explorer 1–2

648

3–10, 12–14

3, 6–14

3, 6–8, 10–14

Describe the likelihood that corresponds to each probability as impossible, unlikely, even chance, likely or certain: a

0.1

50%

70%

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0.6

10%

a pig

a ball or a pig

a pig or an apple

an apple or a star

Consider the spinner to find the probability of getting: A

an apple

a ball

a pig

a star

Find the probability of these events: a

You roll a six-sided die and get 5 or more

An event that will never occur

A favourable event that occurs 2 times out of 50

A favourable event that occurs 2 times out of 100

A favourable event that occurs 2 times out of 500

For each spinner, find the probability of getting a star. Write your answer as a percentage. a

Dylan has 32 marbles in a bag. 20 of them are orange. What is the probability Dylan will pick an orange marble from the bag?

Jack has a bag of 5 marbles, 4 of which are red. Find the probability of Jack picking a red marble, expressed as a decimal.

A single marble is drawn from the jar that has 5 red, 7 blue and 8 black marbles inside.

Find the probability of drawing a blue marble. Give your answer as a percentage.

Find the probability of drawing a blue marble or a red marble. Give your answer as a percentage.

A twelve-sided die is rolled.

Find the probability of rolling 5 or more.

Find the probability of rolling less than 5.

Find the probability of rolling 3 or more.

Find the probability of rolling a number that is divisible by 3.

Find the probability of rolling an odd number.

Ex 2

Consider the spinner to find the probability of spinning:

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Charlene spins this spinner: a

List the sample space.

Find the probability of spinning a number greater than 29.

Find the probability of spinning a number in the twenties.

34 33

A number is randomly selected from this list: {1, 3, 3, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10}

Ex 3

What is the probability of selecting a 1?

What is the probability of selecting a 3?

What is the probability of selecting an 8?

Which number is most likely to be selected?

A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red. a

If the probability of picking a red marble is , how many red marbles are there in the jar?

Find the probability of picking a blue marble as a percentage.

Reasoning

15–17

Paul has 23 marbles in a bag and 4 of them are black. Paul picks a marble from the bag without looking. Find the probability that Paul picks a marble that is not black. Justify your answer.

A bag of marbles contains different coloured marbles. The probabilities of drawing a particular colour are listed: • Red: 10% • Blue: 20% • Green: 30% • Orange: 40% Determine whether the bag could contain marbles of a colour that is not listed. Explain your reasoning.

Consider this spinner. Identify the two equally likely events. Justify your answer with probability.

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Problem-solving 18–19 18

18–19

18–21

A book has pages numbered from 1 to 100. If the book is opened to a random page, what is the probability that the page number: a

is a multiple of 9?

has the digit 6 in the page number?

Draw a spinner that has four outcomes with 2 of them being likely to occur and 2 unlikely to occur.

A card is drawn from a standard deck of cards:

Find the probability of drawing the Jack of Hearts or the Jack of Clubs.

Rosey draws a 6 of Spades and puts it back in the deck and asks you to draw a card. Find the probability that you will draw a card with a number larger than 6 and the same colour as Rosey’s card.

The 26 letters of the alphabet are written on pieces of paper and placed in a bag. If one letter is to be picked out of the bag at random, find the probability of picking: J

K, Y or R.

M, K, D, O, H or B.

A letter in the word PROBABILITY.

A letter in the word WORKBOOK.

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15.03 Theoretical probability After this lesson, you will be able to… • define theoretical probability and differentiate it from relative frequency. • create a sample space for an event. • find the probability of an event using the sample space and express it as a fraction, decimal or percentage. • predict the outcomes of repeated trials of an event based on its theoretical probability.

Theoretical probability We can make predictions for trials by first creating the sample space, then determining the theoretical probability of each outcome.

Let’s look at a theoretical probability example of rolling 2 six-sided dice and adding the numbers together. What is the probability of getting a sum of 6? What about a sum of 10 or greater? 8 or less? Before we answer these questions we need to determine the sample space. The possible outcomes for two dice can be drawn in a grid:

First

1 2 3 4 5 6

1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1

Second 2 3 4 1, 2 1, 3 1, 4 2, 2 2, 3 2, 4 3, 2 3, 3 3, 4 4, 2 4, 3 4, 4 5, 2 5, 3 5, 4 6, 2 6, 3 6, 4

5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5

6 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6

We can now tell that there are 36 possible outcomes. Depending on the event, we can highlight the favourable outcomes corresponding to the event. Since the sample space is 36, we can substitute 36 as the total number of outcomes in the probability formula.

So, Probability (getting a sum of 6) = Probability (getting a sum of 10 or greater) =

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Interactive exploration

Discover this concept in action online

mathspace.co

Example 1 A two-digit number is formed using the digits 3 and 2. It can be two of the same or one of each in any order. a What is the probability that the number formed is odd?

Create a strategy Count the numbers that are odd and use the probability formula.

Apply the idea Four different numbers can be formed: 22, 23, 32 and 33. A two-digit number is odd if its units digit is odd. We have 2 numbers with an odd unit digit out of 4. Substitute the values

Simplify

b What is the probability that the number formed is a multiple of 4?

Create a strategy

Apply the idea

Count the number that can be divided by 4 and use the probability formula.

Four different numbers can be formed: 22, 23, 32 and 33. We have 1 number that is a multiple of 4 out of 4. Substitute the values

Idea summary To find the theoretical probability of an event, use the formula:

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Repeating trials Once we know the probability of an event, we can predict how many times this event will occur if a trial is repeated several times. For example, we know the probability of rolling an even number on a standard die is

since there

are 3 even numbers and 6 total outcomes. If we rolled the die 20 times, how many times would we expect to roll an even number? To answer this, we multiply the probability of the event by the number of trials, rounding to the nearest whole number.

Example 2 An eight-sided die is rolled 25 times. How many times should we expect to roll a 7? Round your answer to the nearest whole number.

Create a strategy Find the probability then multiply it by the number of trials.

Apply the idea The probability of rolling a 7 is . Multiply by the number of trials

Evaluate

Round to a whole number

We can expect to roll a 7 three times.

Idea summary To predict how many times an event will occur if a trial is repeated several times, we multiply the probability of the event by the number of trials.

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15.03 Practice questions Understanding

Adventurer 1, 2bc

Trailblazer 1, 2cd

The probability of event A occuring is . What is the size of the sample space for the trial?

Determine whether these statements are true or false: a

To find the number of expected outcomes, the theoretical probability of event is multiplied to the number of trials.

If a coin is flipped 100 times, it is expected to land on heads 100 times.

If a standard die is rolled 10 times, it is expected to land on every number at least once.

If an event is expected to occur for more than half the trials, then the theoretical probability of the event is likely.

Fluency

3–11

3–7, 9–11

3–5, 7, 9, 11

Yuri has 5 blue marbles, 1 red marble, 1 yellow marble and 1 black marble in a bag. Yuri picks a marble from the bag without looking. Find the probability that Yuri picks a blue marble.

The numbers from 2 to 10 are written on separate cards. One card is chosen at random. Find the probability that the number is: a

Ex 1

Explorer 1, 2ab

a multiple of 2

a prime number

a number less than 6

A three-digit number is formed only using the digits 2, 7 and/or 8. Some of the possible three-digit numbers are listed: 222, 227, 228, 272, 277, 278, 282, 287, 288, 722, 727, 728, 772, 777, 778, 782, 787, 788

Ex 2

List the remaining possible three-digit numbers.

Find the probability that the number formed is odd.

Find the probability that the number formed is palindromic (the same forwards as backwards, such as 101).

If the probability of an event is , how many times would you expect the event to occur in 18 trials?

An eight-sided die is rolled 24 times. How many times should we expect to roll a 7?

This spinner is spun 12 times. Find the number of times we would expect to spin: a

an apple

a ball

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Consider a full set of scrabble tiles where the last two letters are the two “blank tiles”: A A A B C C E E E F G G I I I L M M O O O R R R T T T V V W

A D E G I N O R T W

A D E H I N O R T X

A D E H J N O R T Y

A D E I K N O S U Y

A E E I L N P S U Z

A B E E E F I I L L N O P Q S S U U

One letter tile is drawn at random:

Find the probability that it is a “G” or an “R”.

Find the probability of drawing a vowel.

Luigi and Danielle enter a raffle where 80 tickets are on sale. Luigi buys 1 ticket and Danielle buys 5 tickets. a

Find the probability that Luigi wins if all the tickets were sold.

Find the probability that Danielle wins if all the tickets were sold.

Find the probability that Danielle wins if only 70 tickets were sold.

A twenty-sided die is rolled 100 times. How many times should we expect to roll a 14 or more?

Reasoning

12–13, 15

12–15

A six-sided die is rolled 24 times. How many times should we expect to roll a 1? Explain your process.

If you flip a fair coin 10 times, do you think it will land on heads exactly 5 times? Explain your reasoning.

Consider a game where a standard die is tossed. The rules of the game are: • If a 1 is rolled, you win $2. • If a 6 is rolled, you win $1. • If anything else is rolled, you lose.

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Would you expect to win money or lose the game?

Rowan plays the game once and the die lands on 1. He plays again and the die again lands on 1. Do you think it is possible for Rowan to roll a 1 on his next roll? Explain.

Suppose it costs $1 to play the game. Elsie claims that if she plays the game 3 times, then she is sure to win money. Do you think Elsie should play the game 3 times? Explain your reasoning.

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A die is rolled 358 times. a

If it lands on a six 12 times, what is the probability that the next roll will land on a six?

Does the outcome of the next roll depend on the outcome of previous rolls?

Problem-solving 16 16

16, 18–19

16–20

A bag contains 29 yellow marbles, 21 blue marbles and 10 pink marbles. If a marble is randomly selected from the bag 300 times with replacement, find the number of times that you would expect to pick a marble that is: a

Yellow

Pink

Blue or pink

Yellow, blue or pink

Neil takes these cards and shuffles them up. He draws one card, shows it to you and keeps it. You then draw one card from the remaining seven cards and you win if the number on your card is higher than Neil’s. Find your chance of winning if Neil draws these cards: a

Consider these dice:

Four-sided die

Ten-sided die

Six-sided die

1 2

3 3

Twelve-sided die

Eight-sided die

Twenty-sided die 6

18 12

Identify the die that would give the highest probability of rolling: a

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Amelia selects a card 260 times from a standard deck of 52 cards, she puts the card back into the deck each time before drawing a new one.

How many diamonds can she expect to draw?

How many black cards can she expect to draw?

How many royal cards (Kings, Queens and Jacks) can she expect to draw?

How many times can she expect to draw the King of diamonds?

Sally enters a raffle every week where 130 tickets are sold each week. If there are 52 weeks in a year, find the number of times she can expect to win in a half-year period if she purchases: a

1 ticket every week

10 tickets every week

15.04 Relative frequency After this lesson, you will be able to… • define relative frequency. • calculate relative frequencies. • predict the number of occurrences of an event in a given number of trials by multiplying the relative frequency by the number of trials. • analyse and evaluate experimental data to make informed predictions about future events.

Relative frequency Theoretical probability describes how likely an event is to occur. Relative frequency describes how frequently an event did occur in an experiment.

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To predict the future, we sometimes need to determine the probability by running experiments or looking at data that has already been collected. This is called relative frequency, since we determine the probability of each outcome by looking at past events. Imagine we have a “loaded” die where a weight is placed inside the die, opposite the face that the person cheating wants to come up the most. In this case, the weight is on the bottom because the person wants 6 to come up the most. If the die is made like this, the probability of each outcome is no longer equal and we cannot say that the probability of rolling any particular face is . Instead, we will need to roll the die many times, record our results and use these results to predict the future. Here are the results of an experiment where the die was rolled 200 times. We call this table a frequency table as it shows the frequency of the results. Result 1 2 3 4 5 6

Number of rolls 11 19 18 18 20 114

We can now try to predict the future using this experimental data and the formula:

Here is the table again, with the relative frequency of each face listed as a percentage: Result 1 2 3 4 5 6

Number of rolls 11 19 18 18 20 114

Relative Frequency 5.5% 9.5% 9% 9% 10% 57%

A normal die has around a 17% chance of rolling a 6, but this die rolls a 6 more than half the time. How many of the next 50 rolls can we expect to be a 6? We can calculate this by multiplying the relative frequency of an event by the number of trials. 57% × 50 = 28.5 This rounds to 29 of the next 50 rolls of the die.

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Example 1 A die is rolled 121 times and the results are displayed in the table: Number Frequency

1 18

2 21

3 20

4 21

5 22

6 19

a What is the relative frequency of rolling a 5 with this die?

Create a strategy Use the formula for relative frequency.

Apply the idea We know that the die was rolled 121 times, as it is given above. When we look at number 5 on the table, we can see that it came up 22 times, so we have the:

Reflect and check Notice that if we add up all the numbers in the second column, we get 121, which is the total number of times we rolled the dice.

b What is the relative frequency of rolling a 4 or higher with this die?

Create a strategy Determine how many times the die showed a number 4 or higher then use the formula for relative frequency.

Apply the idea We need to add up the frequency of numbers 4 to 6. This tells us how many times the die rolled 4 or higher. So,

Add the frequencies Evaluate

We already know that the die was rolled 121 times, so we have the:

Example 2 An insurance company found that in the past year, of the 2558 claims made, 1493 of them were from drivers under the age of 25. Give your answers to these questions as percentages, rounded to the nearest whole percent. 660

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a What is the relative frequency that a claim is filed by someone under the age of 25?

Create a strategy Use the formula for relative frequency and convert to a percentage by multiplying by 100%.

Apply the idea Substitute the values

Multiply by 100%

Evaluate

Round to the nearest whole percent

b What is the relative frequency that a claim is filed by someone 25 or older?

Create a strategy Find the numbers of the claims that were made by people who are 25 or older and use the formula for relative frequency.

Apply the idea The number of claims made by drivers 25 or older is 2558 - 1493 = 1065. Substitute the values

Multiply by 100%

Evaluate

Round to the nearest whole percent

Example 3 The relative frequency that a commuter uses public transport is 50%. Out of 500 commuters, how many would you expect to use public transport?

Create a strategy To find the expected number of commuters, multiply the total number of commuters by the probability.

Apply the idea Multiply by a probability

Convert to a fraction

Evaluate

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Idea summary To find the relative frequency of an event we can use the formula:

To predict how many times an event will occur if a trial is repeated several times, we multiply the relative frequency of the event by the number of trials.

15.04 Practice questions Understanding

1–3

Adventurer 1–3

Trailblazer 3

Describe how to calculate the relative frequency of an event occurring.

How is relative frequency different from theoretical probability?

Determine whether these statements describe theoretical probability or relative frequency: a

A coin was flipped 150 times and it was found to land on tails 52% of the time.

Only 1 of the 16 batteries in the package did not work.

When a standard die is rolled, there is a 50% chance of landing on an even number.

Fluency Ex 1

Explorer

5 Ex 2

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4–6, 8, 11–13

5–6, 8, 9, 11–13

The table shows the results of rolling a die: a

How many times was the die rolled?

b c

What is the relative frequency of rolling a 2? What is the relative frequency of rolling an even number?

4, 7, 9, 10, 14, 15

Outcome 1 2 3 4 5 6

Frequency 13 12 20 14 15 16

A retail store served 773 customers in October and there were 44 complaints during that month. Calculate the relative frequency that a customer will complain about the store. Give your answer as a percentage, rounded to the nearest whole percent. Find the relative frequency: a

Flipping a head from a coin that was flipped 184 times with 93 heads recorded.

A computer being faulty at a factory where 1000 computers were tested and 15 were found to be faulty.

A car driving by is white, given that 1919 cars had gone by and 77 of them were white.

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At a set of traffic lights, the green light is on for 24 seconds, then the orange light lasts for 3 seconds and then the red light is on for 13 seconds. This cycle then repeats continuously. If a car approaches the traffic light at a random moment, calculate the probability that the light will be: a

Green

Orange

Red

To prepare for the week ahead, a restaurant keeps a record of the number of each main meal ordered throughout the previous week. a

How many meals were ordered altogether?

Calculate the relative frequency that a customer will order a beef meal.

Meal Chicken Beef Lamb Vegetarian

Frequency 54 32 26 45

Give your answer as a percentage, rounded to the nearest whole percent. 9

11 Ex 3

High school students attending an international conference were asked to register what language they speak other than English. The results are recorded in the table: a

How many students attended the conference?

Find the probability that a student chosen at random speaks:

Language French Arabic Spanish Mandarin Other

French

Mandarin

iii

Arabic or Spanish

Spanish or Other

Frequency 20 13 21 19 37

A factory produces tablet computers. In March it produced 8000 tablets and 240 were found to be faulty. a

Calculate the relative frequency that a tablet produced by the factory is faulty.

The factory plans to produce 9000 tablets in April. How many should they expect to be faulty?

A car manufacturer found that 1 in every 4 cars they were producing had faulty brake systems. If they had already sold 5060 cars, how many of those would they expect to need to be repaired? Uther flipped a coin 14 times. a

Theoretically, how many times would he expect a tail to appear?

After flipping the coin 14 times, Uther noticed that a tail had appeared 4 times. Find the relative frequency of getting a tail.

Derek spun this spinner 20 times. a

How many times would he expect the arrow to land on X?

After he finished spinning, he noticed that the arrow fell on X 8 times. Find the relative frequency of getting an X.

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Five schools compete in a basketball competition. The results from the last season are recorded in the table: Schools competing each game St Trinian’s vs Ackley Bridge Ackley Bridge vs Summer Heights Lakehurst vs Marquess Marquess vs St Trinian’s St Trinian’s vs Summer Heights St Trinian’s vs Lakehurst Ackley Bridge vs Marquess Marquess vs Summer Heights Lakehurst vs Summer Heights Ackley Bridge vs Lakehurst

Winner St Trinian’s Summer Heights Marquess St Trinian’s Summer Heights Lakehurst Ackley Bridge Summer Heights Lakehurst Lakehurst

Calculate the relative frequency that Ackley Bridge wins one of their matches. 15

Georgia is drawing a card out of a deck of 10 cards, labelled from 1 to 10. She drew a card and returned it to the bag 40 times. a

How many times would she expect to get the card with 6 on it?

After she finished, she noticed that the card with 6 on it was drawn out of the bag 21 times. Find the relative frequency of getting a card with 6 on it.

Reasoning 16

16, 18

16–18

Hermione rolled a standard six-sided die 60 times. a

State the theoretical probability of rolling a six.

How many times would she expect a six to appear?

After she finished rolling the die, she noticed that she had rolled a six 24 times. Find the relative frequency of getting a six.

Was the theoretical and relative frequency of getting a six the same? Explain why or why not.

A card is selected at random from a standard deck of cards. The result is recorded and the card is placed back in the deck. This is repeated multiple times. Consider the two tables: Table 2

Table 1 Colour Black Red

Frequency 54 55

Suit Spade Heart Diamond Club

Frequency 21 34 21 33

In which table are the results closest to the expected outcome? Explain your thinking.

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Federica claims that she has created a random number generator which generates a number from 1 to 10. a

She ran the program 10 times and generated this frequency table. Explain whether you think the program is fair or not. Number Frequency

1 1

2 2

3 0

4 0

5 0

1 9

2 11

3 12

4 10

Problem-solving 19, 20

5 8

6 10

8 2

9 2

Calculate the relative frequency that a train will be on time on Monday, as a whole percentage.

Which day had the highest relative frequency of a train being on time?

Calculate the relative frequency of a train arriving on time across the entire week, as a whole percentage.

Boxes of toothpicks are examined and the number of toothpicks in each box is recorded in the table.

Contain 89 toothpicks

Contain more than 90 toothpicks

Contain less than 90 toothpicks

7 11

10 1

8 9

19, 20

The table shows the number of trains that arrived on time at the local station during the week.

If the number of toothpicks of another box were counted, find the relative frequency it will:

7 1

She ran the program 100 times and generated this frequency table. Explain whether you think the program is fair or not. Number Frequency

6 1

10 10

19–21

Day Monday Tuesday Wednesday Thursday Friday

Number of toothpicks 87 88 89 90 91 92 93

The table shows the fighting style of each competitor in a mixed martial arts tournament fought last year.

9 10

Event Karate Wrestling Judo Taekwondo

Number of trains 29 24 21 27 23

On time 22 23 19 22 20

Number of Boxes 0 6 4 1 1 2 1

Frequency 40 54 47 59

Find the total number of competitors.

Find the relative frequency of a wrestler winning.

Find the relative frequency of a Taekwondo fighter not winning.

If 100 more competitors joined the competition, how many of them would you expect to use Karate as their fighting style?

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15.05

Venn diagrams

After this lesson, you will be able to… • interpret the different parts of a Venn diagram. • draw and label a Venn diagram for two or three categories. • use Venn diagrams to solve problems and answer questions related to categories and subsets.

Venn diagrams Venn diagrams provide a visual picture of two sets of data and can be used to represent the sample space in a probability question. Gymnastics

Hockey

In this Venn diagram, the total of the four numbers 5 + 6 + 3 + 15 = 29 represents the total number of people in the survey also called the universal set. The numbers in the shaded area represent the people who play hockey or do gymnastics or do both.

Gymnastics

Hockey

The circle on the left represents students who play hockey, there are 5 + 6 = 11 students.

The circle on the right represents students who do gymnastics, there are 6 + 3 = 9 students. 15

Hockey

Gymnastics

This means there are 15 students who do not participate in either hockey or gymnastics.

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The number outside the circles is the number of students who do not do either sport.

Mathspace WA Curriculum – Year 7 mathspace.co

Hockey

Gymnastics

The middle section highlighted represents the students that are in both categories. This section represents the 6 students who play both hockey and gymnastics.

Hockey

Gymnastics

The part of the circle highlighted is the students who play hockey but do not do gymnastics. 5 students only play hockey.

Example 1 A group of students were asked why they skipped breakfast. The two reasons given were that they were “not hungry” and they were “too busy”.

Not hungry

Too busy

a How many of the students skipped breakfast because they were not hungry?

Create a strategy Find the “Not hungry” circle in the diagram and add the number of students inside it.

Apply the idea Not hungry students = 8 + 11 = 19

Add numbers in the not hungry circle Evaluate

b How many of the students only skipped breakfast because they were too busy?

Create a strategy

Apply the idea

Find the number that is only in the too busy circle.

13 is in the “Too busy” circle and not in the “Not hungry” circle. Too busy students = 13

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c How many of the students skipped breakfast because of one reason?

Create a strategy Add the number of students that are in the “Not hungry” circle or in the “Too busy” circle but not in both.

Apply the idea Students with one reason = 8 + 13 = 21

Add the numbers Evaluate

Example 2 Carl makes a Venn diagram about the possible pets he could get from the local shelter.

Dogs

Fluffy

a Carl is told that a pet will be selected at random. What is the probability that his new pet will be a dog that isn’t ﬂuffy?

9 4

Create a strategy To ﬁnd the probability of a particular outcome we can use the rule:

Apply the idea The “number of favourable outcomes” is the number of dogs that aren’t ﬂuffy. The “total number of outcomes” is the total number of animals who could be selected. So the probability is

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b Carl is told that a pet will be selected at random. What is the probability that his new pet won’t be ﬂuffy?

Apply the idea The “number of favourable outcomes” is the number of animals that are not ﬂuffy. The “total number of outcomes” is the total number of animals that could be selected.

c Carl is told that for a donation they will select his pet from just the dogs at random. What is the probability that his new pet will be ﬂuffy?

Create a strategy Consider that the sample space for this selection is now the dogs and the favourable outcome is a ﬂuffy dog.

Apply the idea The “number of favourable outcomes” is the number of dogs that are ﬂuffy. The “total number of outcomes” is the total number of dogs that could be selected.

Idea summary Venn diagrams provide a visual picture of a problem and can be used to represent the sample space in a probability question.

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Three circle Venn diagrams Venn diagrams can also be drawn with three circles representing three different categories. Comedy

Action

This diagram represents a survey where people were asked if they liked three different movie genres. Horror

This highlighted circle represents people who said they like comedy movies.

This section represents people who said they like comedy, action and horror movies.

This region represents people who said they like comedy and horror, but not action.

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Interactive exploration

Discover this concept in action online

mathspace.co

Venn diagrams don’t always need to be drawn using circles. For scenarios that contain more than 3 categories, it may be beneficial to use rectangles to help visualise the different intersections.

Example 3 Joanne is struggling to decide what to watch online. She decides to pick one movie at random from the streaming website. A Venn diagram of her options sorts movies into three categories based on their genre: Comedy, Action and Horror.

Comedy 10

Action 4

20 6

13 Horror

a How many of the movies are horror ﬁlms?

Create a strategy Add all the numbers inside the “Horror” circle.

Apply the idea Horror films = 13 + 3 + 5 + 6 = 27

Add the numbers Evaluate

b How many of the movies ﬁt into only one genre?

Create a strategy Add the numbers in these parts where the movies ﬁt only one genre:

Apply the idea Number of movies = 10 + 20 + 13 = 43

Add the numbers Evaluate

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c How many of the movies are action films combined with at least one other genre?

Create a strategy Add the numbers in these parts where the movies are action combined with at least one other genre:

Apply the idea Number of movies = 4 + 5 + 6

Add the numbers

= 15

Evaluate

Idea summary Three circle Venn diagrams show how components of three sets are related using three overlapping circles.

15.05 Practice questions Understanding

Adventurer 1, 2

ame the two sets of numbers represented in the N Venn diagram.

List the Multiples of 6 in the diagram.

List the Multiples of 8 in the diagram.

Which number is a multiple of 6 and 8?

Trailblazer 1, 2

Multiples of 6

Multiples of 8

6 12

8 24

18 36

16 32

The Venn diagram shows the brand of phone carried by students in a school. a

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Explorer 1, 2

What does 68 represent?

What does 246 + 68 + 185 represent?

What does 36 represent?

What does 36 + 185 + 68 + 246 represent?

Mathspace WA Curriculum – Year 7 mathspace.co

Sumsang

jPhone 68

246

185 36

Fluency 3

3–8, 10, 12, 14, 15

4–8, 10, 12, 14, 16

A jeweller has arranged his gems based on the cut and colour of the gem:

5, 7, 9, 11, 13, 14, 16–18

Ruby

Triangular

Ruby

Triangular

• Cuts - triangular, rectangular and octagonal • Colours - amethysts, emeralds and rubies

How many of the gems are not rubies?

How many of the gems are both ruby and triangular?

How many of the gems are rubies that are not triangular?

A jeweller has arranged his gems based on the cut and colour of the gem: • Cuts - triangular, rectangular and octagonal • Colours - amethysts, emeralds and rubies

Ex 1

What fraction of the gems are rubies?

What fraction of the gems are triangular?

What fraction of the gems are both rubies and triangular?

A group of students were asked about their siblings. The two categories show if they have at least one brother and if they have at least one sister: a

How many of the students have at least one sibling?

How many of the students have at least one brother?

How many of the students don’t have a sister?

Have a sister

Have a brother

A student creates this diagram of his favourite animals: a

How many of the animals are aquatic?

How many of the animals are aquatic and have teeth?

How many of the animals are aquatic or have teeth, but not both?

Aquatic

Have teeth

Plankton Goldfish Seahorse Jellyfish Dolphin

Dog Tiger

Octopus Earthworm

A group of 60 people were surveyed on whether they enjoyed skateboard riding or bike riding.

Skateboard

Sloth

Bike

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A student is making a Venn diagram from the information gathered from her classmates: 18 people play cricket, 15 play softball and 6 play both softball and cricket. a

How many people only play softball?

Construct a Venn diagram that shows how many students play each sport.

Avril makes a Venn diagram about the possible dance partners in her dance class: a

How many students are there in Avril’s class, not including Avril?

Avril thinks her partner is suitable if they are both good dancers and the right height for her. How many suitable partners are there in the class?

Avril is told that her partner will be randomly assigned. Find the probability that her partner will be suitable.

Play rugby league only.

Play rugby league.

Play rugby union.

Play rugby union only.

Do not play rugby league.

Do not play rugby union.

124

Find the number of students that chose: i

To party.

iii

Neither to study nor party.

Rugby Union 138

169

Party

Study 21

Not to study.

38 10

Find the probability that a student chosen at random chose: i

Not to party.

To study and party.

iii

To study only.

To study or party.

Dan makes a Venn diagram about the possible pets he could get from the local shelter: a

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Rugby League

The Venn diagram shows the number of students choosing to study on the night before an exam and the number of students choosing to party: a

Ex 2

Good dancer

The given Venn diagram shows the number of students in a school playing rugby league, rugby union, both or neither: Find the number of students who:

Right height

Dan is told that a pet will be selected at random. Find the probability that his new pet will be a cat that isn’t ﬂuffy. Dan is told that a pet will be selected at random. Find the probability that his new pet won’t be ﬂuffy.

Mathspace WA Curriculum – Year 7 mathspace.co

Cats

Fluffy

Consider the Venn diagram shown: Find:

P ( A)

P (B)

P ( A)′

P (B)′

P (B ∩ A′)

A 17

Consider the Venn diagram shown: a

P ( A ∩ B)

P ( A ∩ B′)

P ( A ∪ B)

P ( A ∪ B)′

The Venn diagram shows the decisions of 535 consumers choosing to buy an iPhone and consumers choosing to buy a Blackberry: If a consumer is selected at random, ﬁnd the probability that he chose to buy:

Ex 3

19 18

Find:

B 13

A Blackberry.

A Blackberry but not an iPhone.

Both phones.

An iPhone or a Blackberry.

Neither phones.

An iPhone but not a Blackberry.

A 12

How many movies are there in Charlie’s collection?

Charlie wants to watch a comedy. Find the probability he will select a comedy.

19 16

Blackberry

Charlie is struggling to decide what movie to watch. He decides to pick one at random from his collection. A Venn diagram of his collection sorts movies into three categories: comedy ﬁlms, romantic ﬁlms and runtime is over 2 hours: a

B 7

iPhone 68

246

185

Comedy

Romantic

Charlie is after a romantic comedy that goes for less than 2 hours. Find the probability that he will select a suitable ﬁlm.

Charlie is after a comedy ﬁlm that goes for longer than 2 hours. Find the probability that he will select a suitable ﬁlm.

Over 2 hours

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A ﬂorist collected a sample of her ﬂowers and divided them into the appropriate categories as shown in the Venn diagram:

Red flowers

Find the probability that a ﬂower is: a

Not red but has thorns.

Not red and does not have thorns.

Flowers with thorns 14

4 9

10 Flowers with long stems

A group of 94 people were asked about whether they liked cake and/or ice cream. Ice cream

Cake

In total, 76 people said that they liked the cake. 12 people said that they liked ice cream but not cake.

Use this information to ﬁll in all missing values for the Venn diagram.

Reasoning 19

19, 20

19, 21

A student is making a Venn diagram about a recent catch-up test for those who missed or did poorly on the original test. In the class of 30 students, 27 sat for the original test and 10 sat for the catch-up test. a

Given that everyone in the class sat for at least one test, how many people sat for both tests? Explain your answer.

Construct a Venn diagram that shows how many students sat for each test.

A group of people were asked about whether they liked watching basketball and/or baseball. The results are shown in the Venn diagram:

31 Basketball

Baseball 49

Eiichiro makes this claim:

“In total, 70 people like watching basketball and 63 people like watching baseball. When added to the 31 people who do not like watching either sport, this means that 164 people in total were surveyed.” Explain why Eiichiro’s reasoning is incorrect, and ﬁx his mistake(s).

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At a particular school, the languages available for students to study are German, Italian and Spanish. The languages studied by one year group are shown in the Venn diagram.

Five new students joined the school in this year’s group. One of these students studies German, one studies Italian and German, one studies Spanish and Italian and two of them don’t study any languages.

4 16

13 2 12

22–24

A student is arranging a list of seven numbers into two categories. One category is his lucky numbers. Write an appropriate category name for “Circle 1” in the diagram.

9 36 Spanish

Draw a new Venn diagram representing the updated information and explain your reasoning.

Problem-solving 22, 23

Italian

German

22–25

Circle 1

Lucky numbers 31

9 30

A group of people were randomly selected and asked which modes of transport they used to get to work. All people catch either the bus or the train, so no one rides a bike or walks. The Venn diagram shows the results: If one of them is chosen at random, the probability that they catch a bus only is and a train?

Bus

Train

. How many people said they catch a bus

A class of 31 students were surveyed about playing 3 sports. • 16 students play tennis • 14 students play hockey • 2 students play none of these sports • 8 play both tennis and cricket • 7 play both cricket and hockey • 6 play both tennis and hockey • 3 play all three

Construct a Venn diagram for this situation.

Find the probability that a randomly selected student plays all three sports.

Consider the application of a Venn diagram to classify quadrilaterals based on their properties. a

Create a Venn diagram that classiﬁes quadrilaterals by their properties.

Create 3 questions that you could ask a classmate that would require using the Venn diagram to solve.

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MODELLING INVESTIGATION

Recycling probability Explore probability by simulating a rubbish collection at a community soccer field over a weekend. Use a digital simulation to generate counts of plastic bottles, aluminium cans and glass bottles, predict totals for 5 hours and estimate funds raised for recycling.

Objectives • Analyse the rubbish collection scenario and design a simulation to collect data • Calculate probabilities and predict totals using statistical measures and graphs • Interpret predictions and communicate findings in the context of the collection

Analyse the recycling drive You are planning a 5-hour rubbish collection at the local soccer field on the weekend. Simulate a 30-minute sample using the interactive exploration to count plastic bottles, aluminium cans and glass bottles.

Interactive exploration

Discover this concept in action online

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Run the simulation for 30 minutes. Record counts (for example 5 plastic, 3 cans, 2 glass) in a tally chart.

Investigate

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Record the counts from your simulation. Calculate the total items collected.

Why use a simulation instead of physical collection? List one assumption and one limitation.

How might weather or crowd size affect real-world rubbish collection?

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Determine your dataset and analysis Use your simulated sample to calculate the probability of collecting each item type, predict totals for 5 hours of collection and estimate funds raised at 10 cents per item.

Interactive exploration

Discover this concept in action online

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Create a column graph or dot plot showing item probabilities or predicted totals.

Investigate 1.

What is the probability (fraction or percentage) of collecting each item type?

The size of the crowd on the day might be an assumption in the model that affects the probabilities. What other assumptions might be made in this model?

Interpret your plan Evaluate your predictions and their implications for the rubbish collection.

Investigate 1.

Are your predictions realistic for a 5-hour collection? Explain.

Present your collection plan to the class. How do assumptions (for example crowd size) affect it?

Discussion 1.

What challenges did you face in simulating the collection? How did you overcome them?

How could more simulations improve your predictions?

Compare your plan with a classmate’s. How do item probabilities differ and why?

Investigation: Simulations

Investigate online

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15 Chapter review

A standard six-sided die is rolled. Which outcome is least likely? A

Rolling a prime number

Rolling a number less than 3

Describe the likelihood that corresponds to each probability as impossible, unlikely, even chance, likely or certain: a

0.6

80%

A twelve-sided die is rolled.

Find the probability of rolling 6 or more.

Find the probability of rolling less than 6.

Find the probability of rolling a multiple of 4.

12 7

Rolling an even number

8 6

For each spinner, find the probability of getting a star. Write your answer as a percentage. a

A circular spinner is divided into unequal parts. The green sector takes up an angle of 250 degrees at the centre. The red sector takes up an angle of 60 degrees at the centre and the blue sector takes up the remainder of the spinner. What is the probability that the spinner will land on blue?

A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red.

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If the probability of picking a red marble is

Find the probability of picking a blue marble.

, how many red marbles are there in the jar?

Determine whether these events have equally likely outcomes: a

Getting a head or a tail from tossing a fair coin

Drawing a random card from a standard deck of cards

Choosing a random letter from the word MATHEMATICS

Rolling a 1 or 2 on a six-sided die

Getting a random marble from a bag containing red and blue marbles of equal number but of different sizes

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A letter is chosen at random from the word COCOONS. a

Which letter is most likely to be chosen?

Find the probability that the chosen letter is a “C”.

A three-digit number is formed only using the digits 6, 7 and 8. It can be three of the same, two of one and one of another or one of each in any order. a

Find the probability that the number formed is odd.

Find the probability that the number formed is palindromic (the same forwards as backwards).

Consider a full set of scrabble tiles where the last two letters are the two “blank tiles”: A B E F I L O R T V

A C E G I M O R T V

A C E G I M O R T W

A D E G I N O R T W

A D E H I N O R T X

A D E H J N O R T Y

A D E I K N O S U Y

A E E I L N P S U Z

A E E I L N P S U

B E F I L O Q S U

One letter tile is drawn at random: a

Find the probability that it is a “G” or an “O”.

Find the probability of drawing a consonant.

A class of 30 students conducted an experiment where they flipped a coin 10 times each. The total number of heads flipped was 160. Calculate the relative frequency of flipping a head in this experiment.

If the probability of an event is , how many times would you expect the event to occur in 85 trials?

An eight-sided die is rolled 15 times. How many times should we expect to roll a 4?

A die is rolled 670 times. a

If it lands on a six 348 times, what is the probability that the next roll will land on a six?

Does the outcome of the next roll depend on the outcome of previous rolls?

A twenty-sided die is rolled 70 times. How many times should we expect to roll a 12 or more?

Beth enters a raffle every week where 390 tickets are sold each week. If there are 26 weeks in a half-year, find the number of times she can expect to win in a half-year period if she purchases: a

1 ticket every week

9 tickets every week

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An insurance company found that in the past year, of the 2558 claims made, 1493 of them were from drivers under the age of 25. Calculate the relative frequency, as a whole percentage, that a claim is filed by someone: a

Under the age of 25

25 years or older

A retail store served 951 customers in December and there were 26 complaints during that month. Calculate the relative frequency that a customer complains. Give your answer as a percentage, rounded to the nearest whole percent.

The relative frequency that a person uses public transport is 33.33%. Out of 3 people, how many would you expect to use public transport? Round your answer to the nearest whole number.

Hermione rolled a standard six-sided die 60 times.

Determine the theoretical probability of rolling a six.

How many times would she expect a six to appear?

After she finished rolling the die, she noticed that she had rolled a six 24 times. Find the relative frequency of getting a six.

Was the theoretical and relative frequency of getting a six the same? Explain your answer.

To prepare for the week ahead, a restaurant keeps a record of the number of each main meal ordered throughout the previous week. a

How many meals were ordered altogether?

Calculate the relative frequency that a customer will order a chicken meal.

Meal Chicken Beef Lamb Vegetarian

Frequency 52 24 45 36

Give your answer as a percentage, rounded to the nearest whole percent. c 22

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What is the probability of ordering a non-vegetarian meal?

The table shows the number of trains that arrived on time at the local station during the week:

Day Monday Tuesday Wednesday Thursday Friday

Number of trains 23 28 24 27 26

On time 16 26 21 26 21

Calculate the relative frequency that a train will be on time on Monday, as a whole percentage.

Calculate the highest relative frequency of a train being on time out of each of the five days, as a whole percentage.

Calculate the relative frequency of a train arriving on time across the entire week, as a whole percentage.

Mathspace WA Curriculum – Year 7 mathspace.co

The Venn diagram shows the number of students choosing to study on the night before an exam and the number of students choosing to party: a

Find the number of students that chose: i

Not to party.

Both study and party.

Party

Study 21

38 10

Find the probability that a student chosen at random chose: i

Not to study.

To party only.

A jeweller has arranged his gems based on the cut and colour of the gem:

Ruby

Triangular

• Cuts - triangular, rectangular and octagonal • Colours - amethysts, emeralds and rubies

How many of the gems are emerald?

How many of the gems are rubies or triangular but not both?

A student creates this diagram of his favourite animals: a

How many of the animals are not aquatic?

How many of the animals have teeth?

How many of the animals are both aquatic and have teeth?

Have teeth

Aquatic Plankton Goldfish Seahorse Jellyfish Dolphin

Dog Tiger

Octopus Sloth

Earthworm

Consider the Venn diagram shown: Find: a

P (B)

P (A)′

P (B ∪ A′)

P (A′ ∩ B′)

B 13

19 18

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Big ideas Different representations of data highlight different characteristics of the data.

Curriculum Outcomes • Explore and determine the mean, mode, median and range for sets of data and justify, using the context, which measure best reflects the dataset • Represent collected data in a stem and leaf plot, describe the shape and spread including outliers, and compare to dot plots or column graphs. Use the data to estimate probabilities of specific outcomes • Critically analyse statistical statements made in the media and other real-life situations, that relate to the averages of mean, mode and median. Investigate the impact of chance variation on the dataset from which the averages were determined • Analyse, determine and interpret problems in real-world situations that involve assigning a probability to single-stage chance experiments or simulations, statistical measures, stem and leaf plots, dot plots, column graphs and/or Venn diagrams

16 Statistics Chapter outline 16.01 Types of data 16.02 Frequency tables 16.03 Summarise data 16.04 Outliers 16.05 Dot plots and column graphs 16.06 Stem and leaf plots 16.07 Pie charts 16.08 Line graphs Investigation: Statistical investigations Chapter 16 review

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Astronomers use statistics to calculate the probability of finding other planets like Earth.

16.01 Types of data After this lesson, you will be able to… • identify the types of data. • differentiate between categorical data and numerical data. • differentiate between discrete and continuous numerical data.

Types of data Data that is collected as a set of words is called categorical data. Imagine asking someone for their favourite colour, country of birth or gender. Their answer would always be a word. Categorical data is like that - it’s information that can be put into groups or types, based on the words people use to describe things. When the data is a set of numbers, it is called numerical data. Imagine asking someone for their height, their age, how many pets they own or how long they spend on social media each day. Their answer would always be a number. Numerical data is divided into two types: continuous and discrete. Discrete numerical data is counted, so its values are separated and often integer. If you asked someone how many pets they have they might say “1” or “5”; they would not say “10.394”. When you have to count something to know the answer, that’s called discrete data. Continuous data is about things we measure and it can have any value within a certain range. There are so many possible values, we can’t count them all.

5 4

For example, if we measure how tall an animal is, we might find any height that makes sense, but it depends on how accurate our ruler is. When you need to use a ruler or something similar to find the answer, that’s called continuous data.

3 2 1

Example 1 A class was surveyed about where they went on their most recent holiday. What kind of data are the survey results? A Categorical

Discrete numerical

Continuous numerical

Create a strategy Determine whether the answers would be words or numbers. If they would answer with a word, the data is categorical. If they would respond with a number, the data is numerical.

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Apply the idea Names of places are words. So the correct answer is option A.

Example 2 All the students in your school take a survey with the four questions below. Which two will have discrete numerical data as their results? A How many pets do you own? B How many times have you broken your arm? C How long does it take for you to get to school everyday? D What kinds of pet do you own?

Create a strategy Consider the answers to these questions. If the answers are numbers, would they be found from counting (discrete) or would they be found from measuring (continuous)?

Apply the idea Option A requires counting the number of pets. Option B requires counting the number of times they broke their arm. Option C requires measuring time. Option D would be word answers. So options A and B would have discrete numerical data as their results.

Idea summary Categorical data is made up of words. • E xamples include: favourite country, types of pets owned, mode of transport and preferred film genre. Numerical data is made up of two types: • Discrete numerical data is counted. • Examples include: number of pets owned, number of books read, population of a city and goals scored in a season. • Continuous numerical data is measured. • Examples include: Height of a person, weight of an animal, length of a movie and time taken to run a race.

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16.01 Practice questions Understanding

Explorer 1, 2

Adventurer 1, 2

What are the two types of statistical data?

Match the terms to their corresponding definitions: i

Categorical data

Discrete numerical data

iii

Continuous numerical data

These are data that are counted.

These are data that are collected as a set of words.

These are data that are measured and can take any value within a range.

Fluency 3

3, 4a-h, 5a-f

3, 4i-p, 5g-l

Determine whether these data are numerical or categorical: a

Amount owing on lay-by

Types of vegetables

Brands of tablets

Types of folders

Months of the year

Brands of phones

Maximum snowfall

Daily UV index

Types of dogs

Favourite colours

Cost of a PS4 game

Types of fruits

Determine whether these examples of numerical data are discrete or continuous: a

The number of classrooms in your school

The ages of a group of people

Amount of money paid for walking the dog

Lengths of cats’ whiskers

How long it takes you to fall asleep at night

The rainfall in your city

Amount of money owing on a loan

The top speed of different models of motorcycles

Daily humidity

The time taken to run 200 metres

The number of people taller than you in your class

The value of people’s rare coin collections

m The distance from Earth to nearby galaxies

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Trailblazer

The hourly pay rate for a particular job

Number of children in your sports team

The strength of an earthquake

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Ex 1, 2

For each survey question, classify the answer as categorical, discrete numerical or continuous numerical data: a

How far do you travel to school each day?

How many siblings do you have?

How many people live in your house?

What languages do you speak at home?

How long did you spend on homework last night?

How many apps do you have on your phone?

Where was your most recent holiday?

What modes of transport do you use to get to school?

How long does it take for you to get to work each day?

How many times have you ridden your bike?

How tall are you?

Do you think recess is long enough?

Reasoning 6

6, 7

7, 8

8, 9

Determine whether these examples of numerical data are discrete or continuous. Explain your reasoning. a

The amount of soft drink you drink in a day

The number of cans of soft drink you drink in a day

The number of views that a YouTube channel has

The world record time for solving ten Rubik’s cubes

For each survey question, classify the answer as categorical, discrete numerical or continuous numerical data. Explain your reasoning. a

How long did you sleep for last night?

What kinds of pets do you own?

How many pets do you own?

How much does your school bag weigh?

The answer to this question is categorical: “Which students scored higher than 70 in your most recent Science test?” How could you rewrite the question so the answer is numerical? Justify your answer.

The answer to this question is numerical: “How many pairs of shoes do you own?” How could you rewrite the question so the answer is categorical? Justify your answer.

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Problem-solving 10, 12 10

11, 12

11–13

Consider the data set that shows the distance to the nearest hospital (in km): 26.4, 28.7, 15.1, 37.5, 11.8, 7.9, 16.7, 6.9, 5, 24.9, 35.3, 45.5

Is this data set continuous or discrete?

What is the farthest distance to the nearest hospital?

Glen surveys his class and asks, “What is your favourite movie genre?” The results are recorded in the table: Genre Students

Drama 5

Comedy 7

Action 6

Horror 2

What type of data is displayed in the table?

Which genre is the most popular among his class?

How many people did Glen survey?

Ten people were asked about the number of hours they spent outside last weekend. The results of the survey are listed: 5, 9, 9, 4, 6, 9, 4, 9, 6, 4

What type of data is this? If it is numerical, specify if discrete or continuous?

How many people spent exactly 6 hours outside?

What was the longest that anyone spent outside?

What was the most common amount of time spent outside?

Ten people were asked about the number and type of pets they own. The results of the survey are recorded in the table: Twelve fish One dog

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Three turtles One dog, one cat

Two cats One cat

What type of data is the type of pet?

How many cats do these 10 people own altogether?

How many pets do these 10 people own altogether?

What is the most common survey response?

Mathspace WA Curriculum – Year 7 mathspace.co

Two cats Five ferrets

One dog One dog

16.02 Frequency tables After this lesson, you will be able to… • create frequency tables. • interpret data using frequency tables.

Tables, frequencies and modes One common way of collecting data is through a survey. Conducting a survey involves choosing a question to ask and then recording the answer. This is great for collecting information, but at the end we are left with a long list of answers that can be difficult to interpret. A table is a convenient way to organise data, making it easier to analyse and interpret. Frequency refers to the number of times an item or value appears in a dataset. The mode is the value that occurs most frequently in the data set. When analysing data, we can use tables to count the frequency of each value, helping us identify the mode and better understand the distribution of the data.

Example 1 Yvonne asks 15 of her friends what their favourite colour is. She writes down their answer. Here is what she wrote down: Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink a Count the number of each colour and complete the table. Colour

Pink

Green

Blue

Yellow

Number of friends

Create a strategy Count the number of times Yvonne wrote each colour.

Apply the idea Colour Number of friends

Pink

Green

Blue

Yellow

b Which colour is the mode?

Create a strategy

Apply the idea

Choose the colour that occurs the most frequently.

The mode of the data is pink because it occurred the most often.

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Idea summary Tables can be used to organise data and show frequencies, making it easier to analyse and interpret information. The frequency of a result is the number of times that it appears in the list of data. The mode is the value that occurs most frequently in a dataset.

Frequency tables A frequency table is a table that displays the frequencies of different values or categorical data in a data set. It helps summarise data and makes it easier to identify patterns and trends. The word score is also used to mean the values of data. Frequency tables are useful when working with numerical data because they can help us understand the distribution of the data. They can also help answer other questions about the data, such as totals for specific ranges: • To find the total number of data points, add up all the frequencies in the table. • If you want to know the total for “less than” a certain number, add up the frequencies of the results that are smaller than that number. • When finding the total for “at least” a certain number, add up the frequencies that are greater than or equal to that number.

Example 2 Thomas conducted a survey on the average number of hours his classmates exercised per day and displayed his data as shown in the table: No. exercise hours

Frequency

a How many classmates did Thomas survey?

Create a strategy To find the number of classmates surveyed, add all the frequencies of the results.

Apply the idea Classmates surveyed = 2 + 12 + 7 + 5 + 0 + 3 Add all the frequencies = 29

Evaluate

Reflect and check Thomas found that there were no classmates who exercised for 4 hours. Instead of leaving the frequency blank, Thomas put 0 as the frequency. If he had left this information out of the table we would not know how many classmates fit this category.

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b What is the mode of the data?

Create a strategy Choose the result with the highest frequency in the table.

Apply the idea 1 hour of exercise is the mode because it has the highest frequency. c How many classmates exercised for less than three hours?

Create a strategy

Apply the idea

Add the frequencies of the results that represent less than three hours of exercise.

No. classmates = 2 + 12 + 7 Add the frequencies = 21

Evaluate

d How many classmates exercised for at least three hours?

Create a strategy

Apply the idea

Add the frequencies of the results representing three or more hours of exercise.

No. classmates = 5 + 0 + 3 Add the frequencies =8

Evaluate

Idea summary A frequency table communicates the frequency of each result from a set of data.

16.02 Practice questions Understanding 1

Explorer 1, 2

Adventurer 1, 2

Trailblazer 1

Complete each statement using one of the words: i

Frequency

The ⬚ of a data set is the result with the highest frequency.

Mode

iii

Value

The ⬚ of a result is the number of times it appears in the list of data.

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The table shows the students’ responses to their favourite sport. Are these statements true or false? a

Ten students chose cricket as their favourite sport.

Football is the favourite sport of 15 students.

More students chose tennis as their favourite sport than rugby.

Eight students chose swimming as their favourite sport.

Fluency 3

3ac, 4–6, 8, 11

12, 12, 13, 11, 13, 9, 13, 14, 9

14, 2, 5, 2, 2, 5, 5, 7, 12, 13, 13, 14 Score Frequency

21 31

22 37

23 39

24 33

25 38

A gymnast received these scores over 12 rounds of a competition:

3bc, 4, 7, 9, 11

26 34

Score Frequency

Calculate the missing frequency to find how many times the gymnast scored 8. Ex 1

Frequency 10 15 8 12 5

Find the mode(s) of these sets of scores:

3ac, 5, 6, 8, 10

Sport Rugby Football Swimming Cricket Tennis

5 2

6 1

7 1

9 1

Zam asks a number of her friends what their favourite sport is and writes down their responses: Football, Rugby, Football, Swimming, Netball, Rugby, Rugby, Swimming, Netball, Football, Swimming, Rugby, Cricket, Rugby, Cricket

Ex 2

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Construct a frequency table.

Which sport is the mode?

How many friends did Zam survey?

Zack conducted a survey on the average number of hours his classmates played video games per day and displayed his data in the table:

No. of hours 0 1 2 3 4

Frequency 10 12 8 4 2

Price (in cents per litre) 130.0–134.9 135.0–139.9 140.0–144.9 145.0–149.9

Frequency 4 9 12 5

How many classmates did Zack survey?

Find the mode of the data.

How many classmates played video games for 2 hours or less?

How many classmates played video games for at least three hours?

As part of a fuel watch initiative, the price of petrol at a service station was recorded each day for 30 days and displayed in the frequency table:

Mathspace WA Curriculum – Year 7 mathspace.co

What was the maximum price that could have occurred in the 30 day period?

Find the number of days where the price was above 139.9 cents.

The frequency table shows the resting heart rates of a number of people taking part in a study: a

Find the number of people taking part in the study.

How many people had a resting heart rate between 40 and 59 beats/minute?

How many people had a resting heart rate of below 50 beats per minute?

A group of 50 people were surveyed about how many movies they had watched in the past month. The results are recorded in the table: a

Find the number of people who watched 10 or more movies.

Find the number of people who watched less than 5 movies.

Find the number of people who watched more than 4, but less than 15 movies.

Mrs. Tran asks her students how many bedrooms they have in their house and writes their responses:

Number of movies watched 0–4 5–9 10–14 15–19

Number of bedrooms 1 2 3 4 5

3, 3, 4, 4, 3, 3, 4, 5, 2, 2, 4, 4

Heart rate 30–39 40–49 50–59 60–69

Complete the frequency table.

How many students did Mrs. Tran survey?

What is the modal number of bedrooms?

Frequency 20 22 28 37

Frequency 11 14 18 7

Number of students

Bill asks nine of his friends which pet they would like and writes down their responses: Dog, Fish, Cat, Cat, Fish, Dog, Fish, Fish, Dog Construct a frequency table for these responses.

Reasoning 12

12, 13

13, 14

The frequency table shows the number of apples sold by a fruit vendor in a week. Explain how you would find the total number of apples sold. Number of Apples 10 15 20 25

Frequency 5 8 12 6

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A teacher conducted a survey of the favourite colours of 25 students in their class. The results are listed: Red, Red, Red, Red, Red, Red, Blue, Blue, Blue, Blue, Blue, Blue, Blue, Blue, Blue Green, Green, Green, Green, Yellow, Yellow, Yellow, Purple, Purple, Purple

Construct a frequency table based on the given data.

Which colour is the favourite among the students in the class? Explain your answer.

This frequency table shows the occurrence of the vowels in a particular piece of text. Two pieces of data are missing - those for O and U. There are 320 vowel letters on a piece of text. Vowel A E I O U

Frequency 22 28 20

The occurence of O is 1.5 times that of U. Explain the process of determining the two values.

Problem-solving 15, 16 15

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16, 17

Brad stands at the school gate and tallies the hair colour of students as they enter the school. His results are recorded in the table: a

Construct a frequency table from Brad’s tally.

How many students did Brad tally?

What percent of students had red hair? Round your answer to one decimal place.

The table shows the number of trains arriving either on time or late, at Wallarobba station during one week: a

How many trains arrived on time on Thursday?

How many trains arrived late on Wednesday?

What fraction of the total trains arriving during the week, arrived on Friday?

What fraction of the trains arriving on Sunday were on time?

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Hair Colour

Tally

Brown Blonde Black Red

Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Trains on time 14 16 18 15 17 6 5

Trains arriving late 6 4 2 5 3 6 3

A school nurse is investigating the number of sick days taken by students in a class and constructs the table: a

Construct a frequency table for the number of sick days.

What is the number of students who had no sick days?

What is the most common number of sick days for the class?

Student Sally Lachlan Tara Xavier Luigi Nadia Lisa Dave Bill Danielle

Sick days 0 2 1 4 2 4 0 1 3 4

Fifteen airline passengers weigh their luggage. The weights in kilograms are listed: 17, 20, 19, 19, 20, 21, 20, 22, 20, 19, 17, 16, 20, 20, 16 a

Construct a frequency table for this data.

How many passengers had luggage that weighed more than 20 kg?

The airline charges each passenger $3 for every kilogram above 20 kg. Calculate the total charge for overweight luggage for this group of passengers.

16.03 Summarise data After this lesson, you will be able to… • calculate the mean of a data set. • identify the median of a data set. • calculate the range of a data set. • use mean, median and range to solve problems.

Mean The mean is the average of the values in the data set. It is a measure of centre. Three friends are planning a trip to Alice Springs. They plan to fly there and learn that the airline has a rule: each person can only bring 20 kg of stuff in their bags. On the night before the flight they weigh their luggage and find the weights form this data set: 17, 18, 22 One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now?

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Thinking about it using more mathematical language, we are sharing the total luggage equally among three groups. As a mathematical expression, we find:

Each person carries 19 kg. This amount is the mean of the data set. The mean is represented using the symbol . If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same. To calculate the mean, use the formula:

Example 1 Find the mean of these scores: 6, 14, 10, 13, 5, 9, 14, 15 Give your answer as a decimal.

Create a strategy

Apply the idea Use the formula

Add the numbers in the numerator

Perform the division

Idea summary

Mean is the average of the scores.

Median The median is the middle of an ordered set of data. It is a measure of centre. Seven people were asked about their weekly income and their responses form this data set: $300, $400, $400, $430, $470, $490, $2900 The mean of this data set is

= $770, but this amount doesn’t represent the data set very

well. Six out of seven people earn much less than this.

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Instead we can select the median, which is the middle score. We remove the biggest and the smallest scores to get: $400, $400, $430, $470, $490 Then the next biggest and the next smallest to get: $400, $430, $470 Then the next biggest and the next smallest to get: $430 There is only one number left and this is the median - so for this data set the median is $430. This weekly income is much closer to the other scores in the data set and summarises the set better. The median is the number in the middle of a numerical data set. • If the list has an odd amount of numbers, the median is the one right in the centre. • If the list has an even amount of numbers, the median is the number halfway between the two middle ones. Half the numbers in the list will be bigger than the median and half will be smaller. Sets ordered smallest to largest Median Odd number of scores:

Even number of scores:

Median:

+ 2

Example 2 Find the median of these scores: 3, 18, 10, 19, 12, 5, 6, 20, 7

Create a strategy

Apply the idea

We need to put the scores in order and find the middle score.

The scores in order are: 3, 5, 6, 7, 10, 12, 18, 19, 20 The middle score is 10 because it has 4 scores above it and 4 scores below it. The median of the scores is 10.

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Example 3 Two soccer teams, the Lightning Bolts and the Thunder Strikes, have played 5 games each. The scores for each team are listed below: The Lightning Bolts’ scores: 6, 10, 8, 12, 9 The Thunder Strikes’ scores: 5, 8, 11, 9, 12 a Calculate the mean, median and mode for both teams.

Create a strategy We need to put the scores for both teams in order. Use the formula for the mean, find the middle scores and find the scores which occur most often in the data set if applicable.

Apply the idea The scores in order for the Lightning Bolts are: 6, 8, 9, 10, 12 Calculating the mean we have Use the formula

Add the numbers in the numerator

Perform the division

The median is 9 since this is the middle score. There is no mode, as no score occurs more than once. The scores in order for the Thunder Strikes are: 5, 8, 9, 11, 12 Calculating the mean we have Use the formula

Add the numbers in the numerator

Perform the division

The median is 9 since this is the middle score. There is no mode, as no score occurs more than once. b If the Lightning Bolts scored 14 goals in one game instead of 6, how would their mean, median and mode change?

Create a strategy Change one of the scores of the Lightning Bolts from 6 to 14 to find the new mean and new median.

Apply the idea The new scores in order for the Lightning Bolts are: 8, 9, 10, 12, 14

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Calculating the mean we have Use the formula

Add the numbers in the numerator

Perform the division

The mean increased from 9 to 10.6 after changing the score of 6 to 14. This is because the sum of the scores increased, which in turn increased the average. The median increased from 9 to 10, as the new middle score in the sorted list is now 10. There is still no mode, as no score occurs more than once. c If the Thunder Strikes scored 13 goals in one game instead of 11, how would their mean, median and mode change?

Create a strategy Change one of the scores of the Thunder Strikes from 11 to 13 to find the new mean and new median.

Apply the idea The new scores in order for the Thunder Strikes are: 5, 8, 9, 12, 13 Calculating the new mean we have Use the formula

Add the numbers in the numerator

Perform the division

The mean increased slightly from 9 to 9.4 after changing the score of 11 to 13. The sum of the scores increased by 2, which led to a small increase in the average. The median remained the same at 9, as the new score of 13 replaced the score of 11, which wasn’t the middle score in the sorted list. There is still no mode, as no score occurs more than once.

Idea summary To find the median of an ordered numerical data set: • I f the list has an odd amount of numbers, the median is the one right in the centre. • If the list has an even amount of numbers, the median is the number halfway between the two middle ones.

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Range The range is a measure of the spread from the highest score to the lowest. Two bus drivers, Kenji and Bjorn, track how many passengers board their busses each day for a week. Their results are displayed in this table: Kenji Bjorn

M 10 2

T 13 27

W 14 13

T 16 5

F 11 17

Both data sets have the same median and the same mean, but the sets are quite different. To calculate the range, we start by finding the highest and lowest number of passengers for each driver: Kenji Bjorn

Highest 16 27

Lowest 10 2

Now we subtract the lowest from the highest to find the difference, which is the range: Kenji Bjorn

Range 16 − 10 = 6 27 − 2 = 25

Notice how Kenji’s range is quite small, at least compared to Bjorn’s. We might say that Kenji’s route is more predictable and that Bjorn’s route is much more variable. The range of a numerical data set is the difference between the highest and the lowest score. Range = Highest score - Lowest score

Example 4 Find the range of these scores: 10, 7, 2, 14, 13, 15, 11, 4

Create a strategy

Apply the idea

Use the formula Range = Highest score − Lowest score.

The highest score is 15 and the lowest score is 2. Range = 15 − 2 = 13

Subtract 2 from 15 Perform the subtraction

Idea summary

Range = Highest score − Lowest score Range is the difference between the highest and the lowest score.

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16.03 Practice questions Understanding

Explorer 1, 2

Adventurer 1, 2

Trailblazer 2, 3

Describe what the mean measures for a set of values.

Describe what the median measures for a set of values.

A diver measures how long she can hold her breath underwater over several dives. The median time is 3.9 minutes. Are these statements true or false? a

The longest she held her breath was 7.8 minutes.

Most of the time, she held her breath for less than 3.9 minutes.

For half the dives, she was able to hold her breath longer than 3.9 minutes.

The shortest time she held her breath was 1.95 minutes.

Most of the time, she held her breath for longer than 3.9 minutes.

Fluency 4 Ex 1

4a-d, 5, 6a-d, 7, 8, 10–13, 15

4c-f, 5, 6c-f, 7, 8, 11, 12 14, 16

4e-h, 5, 6e-h, 7, 9, 11, 12, 14, 16

Calculate the mean of these data sets: a

8, 5

7, 18, 9, 3

4, 10, 2, 9, 5

16, 4, 7, 2, 5, 19, 6, 11

11, 5, 14, 18, 15

8, 14, 24, 7, 12

13, 27, 9, 6, 8

13, 15, 7, 17, 9

Martha’s grades in her 5 subjects are: 91, 93, 94, 90 and 92 Determine the mean of Martha’s grades.

Ex 2

Ex 3

Determine the median of these data sets: a

1, 4, 6, 8

22, 12, 19, 16

7, 19, 6, 14, 18, 11

6, 9, 3, 5, 4

5, 5, 5, 5

13, 1, 10, 29, 12, 5, 16, 20, 6

5.3, 8.4, 2.6, 4.7, 3.5

49.4, 25.4, 47.2, 34.2, 28.4, 32.2

Two basketball teams, the Red Rockets and the Blue Blazers, have played 5 games each. The scores for each team are: • The Red Rockets’ scores: 75, 82, 88, 90, 85 • The Blue Blazers’ scores: 70, 78, 85, 87, 92 a

Calculate the mean, median and mode for both teams.

If the score for the Blue Blazers first game of 70, was incorrect and they had actually scored 90 how would their mean, median and mode change?

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Two sales representatives, Jacob and Sarah, have recorded their weekly sales (in units) for 5 weeks. The sales for each representative are listed below: • Jacob’s sales: 50, 55, 60, 65, 70 • Sarah’s sales: 45, 60, 55, 60, 65 a b

Calculate the mean, median and mode for both representatives. The next 3 weeks of sales figures come in • Jacob: 40, 35, 38 • Sarah: 45, 48, 50

Does the mean, median and mode go down for both sales agents? 9

Two restaurants, Café Delight and Bistro Bliss, have received customer ratings (out of 10) for 5 weeks. The ratings for each restaurant are: • Café Delight’s ratings: 8, 9, 7, 8, 10 • Bistro Bliss’s ratings: 7, 9, 8, 9, 10

The star ratings look very similar to each other, predict if you think one might have a slightly higher mean than the other?

Calculate the mean, median and mode for both restaurants.

Consider these scores: 3, 6, 8, 10, 2, 11

Ex 4

What is the largest number?

What is the smallest number?

Determine the range.

Calculate the range of these scores: 10, 16, 6, 18, 17, 11, 9, 15, 14

For each data set, determine: i

The mode

Score 11 14 Frequency 10 13

The range 19 4

22 7

29 12

Score Frequency

14 9

19 14

22 12

25 1

How many calls did the employee make during their shift?

For the length of calls made, determine: The mean

The median

iii

The mode

The range

A student scored these grades on his end of year tests: 87, 90, 85, 84, 89, 80, 85, 87, 87 a

How many tests did the student take? i

704

32 10

A phone support centre tracks the length of calls made each day in minutes. During a shift, one employee made calls of these lengths: 2, 3, 3, 2, 2, 4, 4, 2, 5, 13

i 14

The mean

The median

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Determine:

iii

The mode

Which set of data has the smallest range? • Set A: 13, 16, 19, 21, 24 • Set B: 15, 24, 50, 58 • Set C: 6, 8, 16, 35, 40 • Set D: 8, 14, 19, 24, 28

Which set of data has the largest range? • Set A: 101, 105, 118, 129, 136 • Set B: 19, 23, 25, 28, 29 • Set C: 22, 25, 43, 45, 57 • Set D: 104, 107, 113, 128, 141

Reasoning 17

17–19

18, 19

18, 20

Determine whether each statement is always, sometimes or never true. Explain your reasoning. a

The median of a data set is a value in the set.

The mean of a data set is larger than all values in the set.

The range describes how far the lowest data values lies from the highest data value.

Consider this data set that represents the number of apps on six people’s phones: 110, 113, 117, 121, 127, 132 Explain what would happen to the range if each person downloads another 9 apps.

These are the number of points scored by a basketball team in each game of their previous season: 59, 67, 73, 82, 91, 58, 79, 88, 69, 84, 55, 80, 98, 64, 82 a

Determine the range and explain what it represents in context.

Explain why the range might be important for the team’s coach to know.

The television show “Dragon Duels” ran for three seasons, each with 6 episodes. The average fan ratings (out of 10) for each episode are recorded in the table:

Season 1 Season 2 Season 3

Episode 1 8.2 8.2 8.9

Episode 3 7.9 9.1 9.2

Episode 4 7.9 7.8 9.7

Episode 5 8 8.8 7.4

Episode 6 8.2 8.2 4.3

a b

Which season has the greatest mean rating? Which season has the greatest median rating?

If the rating for Episode 6 of Season 3 was 6.3 instead, would your answers to parts (a) and (b) change? Justify your answer.

Problem-solving 21–25 21

Episode 2 8.4 8.6 9.4

21–23, 25–27

22, 23, 25, 26, 28, 29

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A set of 23 scores is arranged in increasing order. In what position does the median score lie?

A set of 34 scores is arranged in increasing order. Between which two scores does the median score lie?

James has played 10 games this season, with a mean of 1.5 goals per game. How many goals did he score in total?

The mean of four values is 21. If three of the values are 17, 3 and 8, what is the fourth value?

In a competition, a contestant must complete 12 challenges earning as many points as possible. Her scores for the first 11 challenges are: 66, 105, 38, 108, 67, 82, 92, 43, 119, 45, 102 Determine her score in the 12th round if the median of all of her 12 scores is 83.5. Counting numbers in increasing order such as 7, 8, 9 are called consecutive numbers.

Identify five consecutive numbers whose median is 6. 28

Identify 3 different sets of four numbers whose median is 12.

Identify four whole numbers whose median is 14 and whose range is 3.

16.04

Outliers

After this lesson, you will be able to… • identify outliers in a given data set. • describe the effect of outliers in the mean, mode, median and range of a data set.

Outliers In statistics, we tend to assume that our data will fit some kind of trend and that most things will fit into a “normal” range. This is why we look at measures of central tendency, such as the mean, median and mode . An outlier is a data point that varies significantly from the body of the data. An outlier will be a value that is either significantly larger or smaller than other observations. There can be more than one outlier.

Interactive exploration

Discover this concept in action online

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Outliers are important to identify as they point to unusual bits of data that may require further investigation. For example, if we had data on the temperature in a volcanic lake and we had a high outlier, it is worth investigating if the sensor is faulty or we need to prepare a nearby town for evacuation. There are formal ways to determine if a data point is an outlier, but for now we are only going to look at data points obviously larger and smaller and how they affect the measures of centre or spread.

Example 1 Identify the outlier in the data set: 63, 67, 71, 76, 111

Create a strategy

Apply the idea

Identify the value that is much greater or much smaller than the other values.

The outlier is 111.

Idea summary An outlier is a data point that varies significantly from the body of the data. An outlier will be a value that is either significantly larger or smaller than other observations.

Effects of outliers Outliers can skew or change the shape of our data. This can be a problem (especially for small data sets) because the mean, median and range might not properly represent the situation. We can counteract this by removing outliers. Usually an outlier will more heavily impact the mean than the median, but each situation needs to be considered using the specific data.

Interactive exploration

Discover this concept in action online

mathspace.co

Removing outliers will have the following effects: Removing a really low outlier The range will decrease. The median might increase. The mean will increase. The mode will not change.

Removing a really high outlier The range will decrease. The median might decrease. The mean will decrease. The mode will not change.

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Example 2 Consider the following set of data: 53, 46, 25, 50, 30, 30, 40, 30, 47, 109 a Find the mean, median, mode and range.

Create a strategy To find the mean, use the formula: To find the median find the middle score, to find the mode find the most frequent score. To find the range, use the formula: Range = Highest score − Lowest score

Apply the idea Use the formula

Evaluate the addition

Evaluate the division

To find the median, order the scores: 25, 30, 30, 30, 40, 46, 47, 50, 53, 109 The middle scores are: 40, 46 Find the average of the middle values

Evaluate

To find the mode, choose the score which occurs most often. Mode = 30 To find the range: Range = 109 − 25 = 84

Substitute the values Evaluate the subtraction

b Which data value is an outlier?

Create a strategy Choose the value that is much greater or much smaller than the rest of the data set.

Apply the idea Outlier = 109

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c Find the mean, median, mode and range after removing the outlier.

Apply the idea Substitute all the scores

Evaluate the addition

Evaluate the division

To find the median, order the scores: 25, 30, 30, 30, 40, 46, 47, 50, 53 The middle score is 40. Median = 40 To find the mode, choose the score which occurs most often. Mode = 30 To find the range: Range = 53 − 25 = 28

Substitute the values Evaluate the subtraction

d Describe how each of the four statistics changed after removing the outlier.

Create a strategy Compare the statistics in part (a) and in part (c) using a comparison table.

Apply the idea Statistics from parts (a) and (c): Mean Median Mode Range

With outlier 46 43 30 84

Without outlier 39 40 30 28

Comparison table: Mean Median Mode Range

With outlier A A A A

> > = >

Without outlier B B B B

The mean, median and range decreased, but the mode stayed the same.

Idea summary Removing outliers will have the following effects on the summary statistics: A really low outlier The range will decrease The median might increase The mean will increase The mode will not change

A really high outlier The range will decrease The median might decrease The mean will decrease The mode will not change

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16.04 Practice questions Understanding

1, 2

Adventurer 1, 2

Trailblazer 2, 3

What is an outlier?

Describe the effect of removing an outlier from a data set on these measures:

Mode

Range

For each scenario, an outlier was removed. Was the outlier smaller or larger than the values that remain? a

The mean decreased after the outlier was removed.

The mean increased after the outlier was removed.

The median decreased after the outlier was removed.

The median increased after the outlier was removed.

Fluency Ex 1

Explorer

710

4, 5, 7, 8, 9a, 10

4, 5, 7, 8, 9c, 10

4, 6–8, 9b

Identify the outlier for each data set: a

9, 10, 12, 9, 1

54, 52, 99, 50, 57

104, 115, 275, 109, 118, 121

4.0, 3.5, 5.1, 2.4, 1.6, 3.9, 3.5, 3.1

Consider the frequency table: a

What is the mode?

Which weight is the outlier?

If the outlier is removed, what is the new mode?

Compare the mode before and after removing the outlier.

Consider the frequency table: a

Which weight is an outlier?

If the outlier is removed, what is the new mean? Round your answer to one decimal place if needed.

If the outlier is removed, is the new mean higher or lower than the current mean?

Mathspace WA Curriculum – Year 7 mathspace.co

Weight in kilograms 14 15 16 17 18 19 20

Frequency 1 0 0 3 6 4 2

Weight in kilograms 12 13 14 15 16 17 18

Frequency 2 5 1 2 0 0 1

For each scenario, determine whether the range would increase, decrease or stay the same: a

If 25 is added to the data set: 1, 1, 2, 3, 3, 3, 4

If 10 is removed from the data set: 10, 42, 55, 60, 65, 70

If 5 and 27 are removed from the data set: 5, 12, 12, 13, 15, 15, 27

If 1 is added to the data set: 24, 25, 25, 26

Consider the data set in the frequency table: Suppose one score of 8 is changed to a 9. Determine whether each measure would change or not:

9 Ex 2

Mean

Median

Mode

Range

Score 1 2 4 5 6 8 9

Frequency 1 2 2 1 2 3 2

For each set of data: i

Find the mean, median, mode and range. Round your answers to two decimal places where necessary.

Which data value is an outlier?

iii

Remove the outlier from the set and recalculate the values found in part (i).

Describe how each of the four statistics changed after removing the outlier.

27, 50, 24, 37, 47, 41, 27, 126, 44, 27

4.7, 2.8, 1.9, 0.9, 0.9, 2.2, 2.2, 1.2, 1.5, 0.9

4700, 4700, 4700, 4500, 5300, 4900, 5200, 4800, 1500, 5100

The weight of fish caught in a “weigh and release” fishing competition, in kilograms are recorded: 12.5, 15.1, 13, 14.2, 14.5, 14.9, 12.5, 14.3, 1.5 a

Find the mean weight.

Find the median weight.

Recalculate the mean weight excluding the outlier.

Recalculate the median weight excluding the outlier.

Reasoning

11, 12, 14

12, 13, 15

The five numbers 16, 16, 17, 24, 17 have a mean of 18 and a median of 17. Describe the effect on the mean and median if a new number is added that is larger than 24.

Carl has been recording his spelling test scores for the past semester. His scores were: 14, 16, 2, 15, 15, 16, 15 a

Calculate the median of Carl’s scores.

Calculate the mean of Carl’s scores, to two decimal places.

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A journalist wanted to report on road speed cameras being used as revenue raisers. She obtained data that showed the number of times 20 speed cameras issued a fine to motorists in one month. The results were: 101, 102, 115, 115, 121, 124, 127, 128, 130, 130, 143, 143, 146, 162, 162, 163, 178, 183, 194, 977 The journalist wants to give the impression that speed cameras are just being used to raise revenue. Which measure of centre should she use in her article? Explain your answer.

The selling prices of artworks sold at an auction are: $18 000, $11 000, $17 000, $20 000, $18 000, $16 000, $15 000, $218 000 Which measure of centre best identifies the typical selling price of recently sold artwork? Explain your answer.

Ted is an employee who works for a company and currently gets paid $26 000 annually. Here is the breakdown of all the employees’ salaries at the company he works for in a frequency table:

Number of employees 4 10 5 3 1

Would the owner of the company use the mean, median or mode to describe the company’s annual pay? Why?

Would Ted use the mean, median or mode to describe the company’s annual pay? Why?

Would a mathematician who is not involved with the company use the mean, median or mode to describe the company’s annual pay? Why?

Problem-solving 16, 17 16

Annual income $26 000 $58 000 $62 000 $150 000 $400 000

16–18

17–19

A group in a study volunteer to take a test that assesses their reaction time. The participants clicked a button as soon as they heard a sound which was played at random intervals. The reaction time, in milliseconds, of each participant is recorded: 220, 280, 210, 220, 215, 180, 185, 190, 190, 195, 150, 190, 195, 195

Identify any outliers.

Which is the most likely explanation for the outlier? A

The participant was not focused.

The sound took longer to play.

The participant missed the button.

The timer malfunctioned.

The selling prices of six houses are: $467 000, $413 000, $410 000 $456 000, $487 000, $929 000

712

What is the median selling price? Round your answer to the nearest thousand dollars.

What is the mean selling price? Round your answer to the nearest thousand dollars.

Which measure of centre most accurately represents the centre of the data?

Which of the selling prices appears to be an outlier?

Recalculate the mean selling price, without the outlier.

Mathspace WA Curriculum – Year 7 mathspace.co

Given the data: 4, 5, 6, 7, 8, x What does x need to be for the mean and median to be equal?

The number of three-pointers scored in different basketball games by a single team are recorded in the table. Describe the effect on each measure if the outlier is removed: a

Mean

Median

Mode

Range

Three-pointers 0 1 2 3 4 5 6 10

Frequency 2 4 5 3 2 2 2 1

16.05 Dot plots and column graphs After this lesson, you will be able to… • create and interpret dot plots, including identifying modes, range and distribution patterns such as positive skew, negative skew, symmetric and uniform. • create and interpret column graphs, focusing on presenting categorical data and understanding the significance of the scale in data interpretation. • identify clusters, gaps and outliers in datasets.

Dot plots The dot plot is a useful way to express discrete data in a visually simple manner. • The main advantages of the dot plot are that we can find the mode and range very easily, as well as quickly see how the data is distributed. • The main disadvantages are that we need to count each dot when finding the median and it is often easier to convert to a table to find the mean. • The dot plot is particularly suited to discrete data where the frequency of results are often greater than one. In a dot plot, each dot represents one data point belonging to the result that it is placed above. The modes of a dot plot will be the results with the most dots. Since a dot plot stacks vertically, the highest columns will belong to the modes.

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Sometimes we can describe the data in a dot plot using the shape of the plot. When the dot plot has a negative skew, the tail of the data is to the left on the plot. When the dot plot has a positive skew, the tail of the data is to the right on the plot. Other distribution patterns include symmetric and uniform: Direction of tail

Direction of tail

0 1

2 3 4 5 6 7 8

9 10

Negative Skew

The dot plot shows a negatively skewed distribution.

80 85 Symmetric

The dot plot shows a positively skewed distribution.

10 70

3 4 5 6 7 Positive Skew

Uniform

The dot plot shows a symmetrical distribution.

The dot plot shows a uniform distribution.

Clusters on dot plots are groups of dots that are close together, showing that many data points have similar values. Gaps are spaces with no dots, which means there aren’t any data points with values in that range.

Example 1 A group of adults is asked: “How old were you when you passed your driving test?”. The responses were: 22, 17, 17, 17, 19, 21, 17, 22, 21, 18, 18, 17, 18, 22, 18 This dot plot represents the responses. Responses

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a What is the range of this data set?

Create a strategy To find the range, use the formula: Range = Highest score − Lowest score

Apply the idea Range = 22 − 17 =5

Find the difference Evaluate the subtraction

b What is the mode of this data set?

Create a strategy

Apply the idea

Choose the score which has the most dots above it.

Mode = 17

c What is the median of this data set?

Create a strategy

Apply the idea

Order the scores and find the middle score.

Arrange the scores in order: 17, 17, 17, 17, 17, 18, 18, 18, 18, 19, 21, 21, 22, 22, 22 The middle score is: 18 Median = 18

d How many people passed their driving test on or after their 19th birthday?

Create a strategy Count the number of dots that are greater or equal to 19.

Apply the idea No. People = 6

Count the scores

Example 2 Sarah celebrated her 13th birthday at a bowling alley. She invited 20 friends and they played a game of bowling. The scores for the game were: 12, 17, 23, 31, 35, 42, 45, 49, 49, 49, 49, 53, 56, 65, 69, 75, 75, 83, 83, 300 This dot plot represents the scores.

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

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a Describe the shape of the data.

Create a strategy

Apply the idea

Check where the bulk of the data sits and look at the general shape of the distribution.

The scores have higher-frequency columns on the left (the lower scores), so the distribution is positively skewed. There appears to be an outlier as there is one value far to the right of the others. The data set also shows a large gap between 83 and 300 and clustering around 50.

b Determine the mean and median of the data.

Create a strategy

Apply the idea Add all the 20 scores

To find the mean, use the formula: To find the median, find the middle score.

Evaluate the division

Since the number of scores is even, then the two middle scores are 49 and 49, then this means that: Median = 49

c If we remove the outlier, how does the shape of the data change?

Create a strategy

Apply the idea

Check where the bulk of the data sits after removing the outlier 300 and look at the new general shape of the distribution.

The distribution will become less skewed and more symmetric, as the outlier was pulling the mean towards the higher end of the scale. The median will remain unchanged, as it is not affected by the outlier.

d State the mean and median of the data without the outlier.

Create a strategy

Apply the idea

To find the mean, use the formula:

Add all the 19 scores

To find the median, find the middle score.

Evaluate the division

Since the number of scores is odd, then the middle score is 49. Median = 49

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Example 3 Consider the dot plot:

9 10 11 12 13 14 15 16 17 18 19

a Where does the clustering occur?

Create a strategy

Apply the idea

Identify the particular range on which the group The cluster of scores occurs between 15 and 19. of dots are close together or ‘bunched’ together.

b Determine the location of any gaps in the data.

Create a strategy

Apply the idea

Identify the particular range on which there are spaces with no dots.

There are gaps between the scores of 6 to 8 and 13 to 14.

Idea summary In a dot plot, each dot represents one data point belonging to the result that it is placed above. The modes of a dot plot will be the results with the most dots. Since a dot plot stacks vertically, the column(s) with the most dots will belong to the mode(s). The data in a dot plot can be described based on the shape of the distribution. We can use words like negative skew, positive skew, symmetric, uniform, clusters and gaps to describe what we see on a dot plot.

Column graphs A column graph always presents a categorical data set. There is one column per category and the height of each column is the size of that category. There is usually a small gap between each column. The scale of a column graph (indicated by the numbers and ticks on the vertical axis of the graph) is really helpful when looking at data, so we should know how to read it. Think of the scale like a number line that goes up. The marked numbers show the value at different heights and the little marks in between help us figure out the values at other heights. The scale of a column graph is important for reading values, however there are some cases where the scale may be misleading.

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In this column graph, if we look only at the columns and ignore the scale, it certainly seems like the number of albums sold doubled from 2014 to 2015. However, if we look at the scale, we can see that the number of albums sold in 2014 and 2015 were 25 thousand and 30 thousand respectively, so the sales did not double.

Number of albums (in thousands)

Albums sold per year 40 35 30

The reason why the heights of the columns are so misleading is because the scale doesn’t start at zero on this graph. It starts at 20 thousand, so the height of each column indicates how much greater than 20 thousand the value is.

25 20

2012 2013 2014 2015 2016 Year

We should always check the scale to find actual values before making any conclusions about the data.

Example 4 A column graph is presented, showing the type of fruit that each student in a class brought to school for lunch yesterday.

Fruit for Lunch

10 9 8 Number

7 6 5 4 3 2 1 0

Banana

Apple

Mandarin Pineapple Orange Fruit

a What was the most common fruit? A

Apple

Mandarin

Orange

Banana

Pineapple

Create a strategy

Apply the idea

Check the graph to see which fruit has the tallest column.

Mandarin has the tallest column, so the correct answer is option E.

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b How many more mandarins were brought to school than pineapples?

Create a strategy

Apply the idea

Check the graph to see how many pineapples and mandarins were brought to school, then find the difference.

The number of pineapples is 2 and number of mandarin is 8. Difference = 8 − 2 =6

Subtract the 2 from 8 Evaluate

c Complete the table using the information from the graph. Fruit Banana Apple Mandarin Pineapple Orange

Number 2 ⬚ ⬚ ⬚ ⬚

Create a strategy

Apply the idea

Remember, the height of each column represents the number of each fruit that was brought to school.

By checking the height of each column, we can fill in the numbers of each fruit. Fruit Banana Apple Mandarin Pineapple Orange

Number 2 3 8 2 4

Example 5 The table shows the number of people who visited Disneyland between 2011 and 2015. Year 2011 2012 2013 2014 2015

Number of people in hundreds of thousands 165 164 152 159 168

a Represent the data in a column graph.

Create a strategy We assign the values from the table for each year to make the column graph. The Year should go on the horizontal axis and the Number of people should go on the vertical axis.

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Apply the idea Number of people (in hundred-thousands)

Disneyland Visitors 170

165

160

155

2011 2012 2013 2014 2015 Year

Note that the vertical axis does not start at zero. b A marketing executive examines the column graph and says, “We doubled the number of visitors from 2014 to 2015.” Are they correct?

Create a strategy

Apply the idea

Read the values for each column from the graph. The column for 2014 goes up to 159 thousand and the column of 2015 goes up to 168 thousand which is not double 159 thousand. So the executive is not correct.

Idea summary A column graph always presents categorical data. We should always check the scale to find actual values before making any conclusions about the data.

16.05 Practice questions Understanding

Explorer 1–3

Adventurer 1–3

How can you determine the frequency of a category on a dot plot?

What does the mode look like on a dot plot or column graph?

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Trailblazer 1–3

If the frequency of each category in a set of data is high, is it better to use a dot plot or a column graph to represent the data?

Fluency 4

4, 5, 7–10, 13, 14, 16–19

9, 17–20

A group of adults are asked: “How old are your kids?” The responses were: 15, 4, 18, 15, 9, 10, 5, 15, 16, 5, 16, 17, 10, 12, 19 a

Plot the responses on a dot plot.

Determine: i

c 5

the range

the mode

iii

Sophia is a casual nurse. She used a dot plot to keep track of the number of shifts she did each week for a number of weeks: a

In this dot plot, what does each dot represent?

How many weeks did she work 5 shifts?

What was the most frequently occurring number of shifts per week?

Describe the shape of the distribution of the number of shifts per week.

Consider the column graph showing the pets that the children in a class have: a

Snakes are the least popular pet.

Two children have dogs as their pets.

Turtles are the second most popular pet.

No child has a cat as a pet.

Number

Are these statements true or false?

5 6 7 Shifts per week

Pets

10 9 8 7 6 5 4 3 2 1 0 Dog

Snake

Crab

Cat

Turtle

Animal

These graphs show the annual net profit (in millions) of a company over a several year period. For each graph, determine the year in which the annual net profit is an outlier: a

Annual Profit 80 70 60 50 40 30 20 10 0

120 Net Profit (millions)

the median

How many people have kids that are 15 or older?

Net Profit (millions)

Ex 1

5, 6, 8–12, 14–17

2005 2006 2007 2008 2009 2010 Year

Annual Profit

100 80 60 40 20 0

2004 2005 2006 2007 2008 2009 Year

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Glen wants to put the information from this column graph into a table:

Favourite Sports Number of People

5 4 3 2 1 0

Cricket Basketball Soccer

Tennis

Netball

Sport

Complete this frequency table to match the information in the column graph.

Ex 2

Sport Cricket Basketball Soccer Tennis Netball

Number of People

During a school field trip, a group of students participated in a laser tag game. The scores for the game were: 5, 10, 15, 20, 25, 30, 35, 40, 40, 40, 40, 45, 50, 55, 60, 60, 65, 70, 75, 200 This dot plot represents the scores.

Ex 3

100

120

140

160

Describe the shape of the data.

Determine the mean and the median of the data.

If we remove the outlier, how does the shape of the data change?

Determine the mean and median of the data without the outlier.

200

A group of people were surveyed on the number of websites they visit on a daily basis. The results are displayed in the dot plot: Number of websites visited daily

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Complete the frequency table based on the dot plot. Number of websites visited daily 0 1 2 3 4 5 6

Frequency

Identify any clusters in the number of websites visited daily.

Are there any gaps in the distribution of websites visited daily? If so, between which values?

This dot plot shows the number of students in each section of the school band: a

Which section has the least members?

Determine the number of people in the band.

How many more people are there in the Strings section than in the Brass section?

Describe the shape of the distribution of students in the band sections.

Band members

Strings

Woodwind

Each judge on a cooking show gives contestants a rating between 1 and 5. The dot plot shows the scores given to the winning contestant: a

Describe the shape of the distribution of ratings given by the judges to the winning contestant.

How many judges gave the contestant a rating of 4?

How many judges gave the contestant a rating of 4 or more?

How many judges gave the contestant a rating of 2 or less?

Contestant rating

The goals scored by a football team in a single season are represented in the dot plot: a

What is the modal number of goals scored?

Determine the median number of goals scored.

Brass Percussion

Goal scores

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In a study, a group of people were shown 30 names and after one minute they were asked to list the names they could remember. The results are presented in the dot plot: a

How many people took part in the study?

What is the highest number of names someone remembered?

What was the lowest number of names someone remembered?

Determine the range of names remembered.

A student conducts a survey asking their class members what their favourite video game is. The results are displayed in this column graph: a

Which is the most popular game?

Which two games were equally popular?

How many students took part in the survey?

Names Remembered

Favourite Video Game 10 9 8 7

Frequency

6 5 4 3 2 1 0

Blobrox

Underlook

Bash Sisters

Spawn Campers

Elf Fortress

Game

This column graph shows the star ratings by different patrons at a local hotel: a

What is the mode?

How many ratings of 5 were given?

How many more people gave a rating of 3 than a rating of 2?

Frequency

Ex 4

Construct a frequency table using the information from the graph. 1

2 3 4 Hotel Ratings

Yvette has 7 black marbles, 3 blue marbles, 9 yellow marbles, 12 white marbles and 5 red marbles. Construct a column graph to represent this information.

Ex 5

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The table shows the number of people (in millions) who visited the Sydney Opera House between 2015 and 2019: a

Use the information from the table to construct a column graph.

A marketing executive examines the graph and says, “There has been a significant increase in the number of visitors from 2015 to 2019”. Is this statement correct?

Mathspace WA Curriculum – Year 7 mathspace.co

Year 2015 2016 2017 2018 2019

Number of people in millions 8.2 8.8 9.6 10.0 10.8

The average monthly attendance of a fitness class is recorded in the table: Attendance

August 42

September 41

October 43

November 39

Construct a column graph to represent the given data.

In which month is the average attendance at a minimum?

In which month is the average attendance at a maximum?

Determine the range.

December 40

Aviation glass is manufactured to be 12 mm thick. If the glass is too thin (less than 11.5 mm) or too thick (more than 12.5 mm) the aircraft will fail its glass safety check. A regional airport conducts a glass safety check of all aircraft landing on their runways over a one month period and displays the results in the dot plot: a

How many aircraft were checked for glass safety?

What is the lowest level of glass thickness measured?

What is the modal glass thickness measured?

What percentage of aircraft failed the glass safety check?

Reasoning 21

21, 23

Glass thickness (in mm)

10.5

11.0

11.5

21, 22a, 23

12.0

12.5

13.0

21–23

The dot plot shows the temperature (°C) in a town over several weeks.

Temperature in a town

Which measure of centre would represent this data best? Explain your reasoning.

Consider the column graph. a

Darius says the median score is 48.5 since 48 and 49 are the two scores in middle. Is he correct? Explain your reasoning.

Explain a method for calculating the mean score.

Frequency

10 9 8 7 6 5 4 3 2 1 46 47 48 49 50 51 Scores

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The table shows the house points earned by four houses at their swimming carnival: a

Construct a column graph using the information from the table.

Would it be reasonable to say that Dundalli had twice as many points as Nemarluk? Explain your reasoning.

Problem-solving 25 24

726

House Truganini Nemarluk Dundalli Umbarra

24, 25

Points 40 30 70 50

24–26

A supermarket manager takes a note every time an employee is late to work and how late they were (rounded to the nearest half hour). The dot plot shows their results for the last month:

Lateness (in hours)

Determine the median amount of hours that employees were late by.

What fraction of late employees were later than the median amount?

If an employee is more than 1 hour late, the manager fines them $10. How much money did the manager collect in fines over the last month?

These dot plots display how many seconds it took for students to race from their classroom to their locker: Mr. Navra’s Class

Mr. Lee’s Class

50 51 52 53 54 55 56 57 58 59 60

What is the slowest time recorded?

Which teacher’s class had the largest range of times?

Which teacher’s class had the fastest student?

Which teacher’s class had the slowest median time?

In each game of the season, a basketball team recorded the number of ‘three-point shots’ they scored. The results for the season are represented in the dot plot:

How many games did they play during the season?

If the team scored 4 three-point shots in a game, how many points are scored from three-point shots?

What was the total number of points they scored from three-point shots during the season?

Determine the mean number of points they scored per game from three-point shots, rounded to two decimal places.

Mathspace WA Curriculum – Year 7 mathspace.co

Number of Three Points Shots

16.06 Stem and leaf plots After this lesson, you will be able to… • create stem and leaf plots. • interpret data presented in stem and leaf plots. • identify distribution, gaps and clusters in split stem and leaf plots.

Stem and leaf plots A stem and leaf plot is a type of data display that organises data points into stems (representing ranges of values) and leaves (representing individual values within those ranges). We use stem and leaf plots when we have many numerical data points. Stem Leaf 5 2 Key: 5|2 = 52

The score 52 is expressed on this stem and leaf plot. The ones digit is in the row corresponding to its tens digit. In other words, we attached the leaf, 2, to its stem, 5, to make the score 52.

What is useful about the stem and leaf plot is that we can record as many scores as we like by writing the leaves in the appropriate rows. Stem Leaf 3 0 1 4 6 9 5 1 2 2 7 Key: 5|2 = 52

We can express the data set: 52, 46, 31, 57, 49, 51, 52, 30 with this stem and leaf plot. As we can see, each score has been expressed on the plot as a ones digit written in its tens row and the values in the leaf column are written in ascending order leading away from the stem.

While stem and leaf plots are used primarily to store data of two digit numbers, there are some cases where the stem and leaf might mean something different. It is for this reason that we should always check the key before translating the leaves into scores. Stem Leaf 1 3 6 7 2 0 2 2 7 3 8 9 Key: 2|7 = 2.7 km

For example, in this stem and leaf plot the stem represents the whole number of kilometres while the leaf represents tenths of a kilometre.

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There are also cases of the stem representing the number of tens as usual, except it uses two digit numbers in the stem to express three digit numbers. Stem Leaf 9 2 5 10 0 3 3 9 11 7 8 12 3 4 6 8 13 1 1 4 Key: 12|8 = 128

In this case, the score 128 is represented by the leaf “8” attached to the “12” stem.

A split stem and leaf plot is a variation that divides each stem value into two or more parts for a more detailed representation. To read both types of plots, look at the stem values on the left side and match them with the leaf values on the right side, while considering the divisions within each stem for split plots. The stem and leaf plots shown contain the same data: Stem Leaf 3 0 1 3 6 4 6 9 5 1 2 5 5 6 6 6 6 7 7 8 8 9 Key: 5|2 = 52

Stem 3 3 4 4 5 5

Leaf 0 1 3 6 6 9 1 2 5 5 6 6 6 6 7 7 8 8 9 Key: 5|2 = 52

A split stem and leaf plot provides a more precise visual representation of the data, allowing for better identification of distribution, gaps, clusters and facilitating comparisons with other data sets. Consider how the split stem and leaf plot of the same data might help us identify these features.

Example 1 A city council selected a number of houses at random. They determined the fastest travel time (in minutes) from each house to the nearest hospital and recorded the following results: 25, 37, 16, 27, 27, 35, 21, 18, 19, 49, 14, 19, 31, 42, 18 Represent this data as an ordered stem and leaf plot and comment on the distribution.

Create a strategy Put the scores in a stem and leave plot where the tens digit is the stem and the leaves are in increasing order.

Apply the idea Stem 1 2 3 4

Leaf 4 6 8 8 9 9 1 5 7 7 1 5 7 2 9 Key: 2|5 = 25

From the stem and leaf plot we can see that the data has a slight positive skew with no clusters or gaps.

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Example 2 The following stem and leaf plot shows the ages of 20 employees in a company. a How many of the employees are in their 30s? Stem Leaf 2 0 1 1 2 8 8 9 3 0 2 4 8 8 4 1 1 1 2 5 5 3 4 8 Key: 1|2 = 12

Create a strategy

Apply the idea

Count how many leaves the stem 3 has.

Employees = 5

b What is the age of the oldest employee?

Create a strategy

Apply the idea

Find the largest leaf of the largest stem and put them together to find the oldest age.

Age = 58

c What is the age of the youngest employee?

Create a strategy

Apply the idea

Find the smallest leaf of the smallest stem and put them together to find the youngest age.

Age = 20

d What is the median age of the employees?

Create a strategy

Apply the idea

Since there are 20 scores, find the average of the 10th and 11th score.

By counting the leaves, the 10th score is 34 and the 11th score is 38. Find the average of the scores Perform the addition Perform the division

e What is the modal age group? A 20s

30s

40s

50s

Create a strategy

Apply the idea

The modal age group is the most common age group.

Stem 2, which represents the 20s, has the most leaves. So the correct answer is option A.

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Example 3 Consider the stem and leaf plot depicting the various amounts of time a group of students spend each day reading:

Stem 0 1 2 3 4

Leaf 4 0 1 6 6 7 0 2 3 3 3 3 4 6 8 1 1 1 2 5 6 6 8 3 4 Key: 3|5 = 35 minutes

a Represent this data as a split stem and leaf plot.

Create a strategy For each group of tens (such as 0, 10, 20, etc.), write two stems in a vertical list. One stem is for numbers ending in 0 – 4 and the other is for numbers ending in 5 – 9.

Apply the idea Stem Leaf 0 4 0 1 0 1 1 6 6 7 2 0 2 3 3 3 3 4 2 6 8 3 1 1 1 2 3 5 6 6 8 4 3 4 4 Key: 3|5 = 35 minutes

b Compare the shape, centre and spread of the split stem and leaf plot to the stem and leaf plot, including the mean and median.

Create a strategy Consider whether one of the plots shows if the data is skewed, determine where the centre of the data lies and how the visual representation of each set shows that and if there are any clusters or outliers in the data.

Apply the idea • The stem and leaf plot shows a symmetric distribution of the times, while the split stem and leaf plot shows that there is more of a positive skew in the data. • The mean of the data is 27.68 and the median is 24. Comparing both plots, the split stem and leaf plot shows that a cluster of students reads between 20 and 24 minutes each day. We can use the split stem and leaf plot to give a better visual of the centre. • The 4-minute data point appears to be an outlier when the data set is placed in a split stem and leaf plot.

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Idea summary A stem and leaf plot is made up of two components, the stem and the leaf. The stem is usually used to represent the tens part of a score while the leaf is used to represent the ones part of the score. A split stem and leaf plot is a variation that divides each stem value into two or more parts for a more detailed representation.

16.06 Practice questions Understanding 1

Trailblazer

1–3

2, 3

Stem

The stem and leaf plot is made up of two components, the ⬚ and the ⬚.

1–3

Adventurer

Complete each statement with one of these words:

Explorer

Leaf

The ⬚ is used to represent the ones part of the score

The ⬚ is usually used to represent the tens part of a score.

Consider the stem and leaf plot: a

Using the given key, what number does 3|0 represent?

If the data value 65 is added to the stem and leaf plot, which digit is the stem and which is the leaf?

What is the smallest value?

What is the largest value?

Stem 1 2 3 4 5

Leaf 2 8 8 9 2 2 3 4 7 8 0 2 8 1 0 2 Key: 1|2 = 12

Are these statements about stem and leaf plots true or false? a

The data values are ordered.

A stem and leaf plot does not show outliers clearly.

A stem and leaf plot is only appropriate for data where values have high frequencies.

The individual data values can be read on a stem and leaf plot.

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Fluency 4

4–6, 8–10

4–10

This stem and leaf plot records the ages of customers at a beachside cafe last Sunday. Use the stem and leaf plot to complete the frequency table: Stem Leaf 1 1 5 9 2 2 3 5 6 7 9 3 0 1 2 4 6 7 4 1 3 4 6 8 9 5 1 4 7 6 2 4 5 6 7 Key: 5|2 = 52 years old

The stem and leaf plot shows the number of hours worked per week by a group of people: Identify the outlier.

Ex 1

6–8, 10–11

Age 10−19 20−29 30−39 40−49 50−59 60−69

Frequency

Number of hours Stem Leaf 1 2 2 7 7 9 9 3 1 3 3 3 3 5 8 4 4 4 5 Key: 4|5 = 45 hours

The data shows the prices paid for a ticket to a football match: 94, 66, 73, 96, 75, 60, 76, 84, 77, 82, 104, 84, 67, 85, 87, 91, 76, 97, 99, 64 Represent this data as an ordered stem and leaf plot.

Consider the stem and leaf plot: Find the mean of this data set.

Consider the stem and leaf plot: Find the median of this data set.

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Stem Leaf 1 2 6 8 9 2 0 2 9 3 0 2 Key: 1|2 = 12

Stem Leaf 1 2 9 2 4 6 7 3 0 4 5 6 4 3 5 7 9 5 0 2 6 5 7 3 9 8 4 5 Key: 5|2 = 52

Ex 2

11 Ex 3

An online booking website allows customers to buy tickets to concerts. Over the last month, the website has been tracking how many tickets are sold in the first minute after they go on sale. The results are displayed in the stem and leaf plot: a

How many concerts does this data set represent?

Find the highest number of tickets sold for a concert in the first minute.

Find the range of the data.

The stem and leaf plot shows the age of people who enter through the gates of a theme park in the first ten minutes: a

How many people passed through the gates in the first ten minutes?

What is the age of the youngest person?

What is the age of the oldest person?

Find the median age.

Find the modal age.

Roxanne is using a precision telescope to measure the size of craters, in kilometres, closest to Olympus Mons on the surface of Mars. She records her findings in this stem and leaf plot:

Stem Leaf 1 0 0 1 2 2 3 4 8 2 0 1 5 7 8 9 3 0 2 6 8 4 5 6 Key: 1|2 = 12 tickets sold

Stem 1 2 3 4 5

Stem 1 2 3 4

Leaf 1 2 3 3 4 4 6 6 7 0 2 4 9 9 2 5 5 5 3 Key: 1|2 = 12 years old

Leaf 1 2 3 6 7 8 8 1 3 4 5 5 5 0 1 1 2 3 4 4 5 7 8 8 0 Key: 1|2 = 12 kilometres

Find the median crater size.

Find the range of crater sizes.

Roxanne finds that the sum of all the crater sizes is 661 km. Find the mean crater size, rounded to two decimal places.

Represent the stem and leaf plot as a split stem and leaf plot.

Reasoning

12, 13, 14a, 15

13–15

12–15

Explain how can you use stem and leaf plots to identify the range of a given data set.

Consider this stem and leaf plot: Which measure of centre best represents the centre of the data? Explain your reasoning.

Stem Leaf 2 3 3 4 5 6 7 1 3 5 8 8 1 3 3 4 5 9 3 4 Key: 1|2 = 12

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The data represents how long students in two different classes could hold their breath for, measured to the nearest second. Ms. Jacobs’ Class Stem Leaf 1 5 5 2 9 3 3 6 2 7 4 7 5 5 6 5 0 6 4 5 7

Mr. Ross’ Class Stem Leaf 1 5 0 7 2 6 4 3 3 5 4 7 6 3 4 5 3 7 6 1 9

If you want to know which class, in general, is better at holding their breath, which measure of centre would be best? Explain your reasoning.

The stem and leaf plot shows that the lightest parcel arrived in a courier service is 11 kg while the heaviest is 53 kg.

The median mass is 2.2 kg.

16, 17

The level of mercury in 40 different water samples was tested and recorded in the stem and leaf plot:

Stem 9 10 11 12

Stem 1 2 3 4 5

1 6 6 8 0

Leaf 2 2 4 4 2 3 9 3

16–19

Leaf 3 3 3 3 6 7 8 8 0 0 3 3 4 4 5 5 2 3 4 4 5 6 8 9 0 1 1 1 2 2 3 5 Key: 11|2 = 112 units

9 9 6 7 9 9 6 7

What is the highest reading?

Find the 8th lowest reading.

How many readings of 100 or higher were recorded?

The level of mercury is deemed to be safe if the reading is 103 or lower. What fraction of the readings are safe?

Represent the stem and leaf plot as a split stem and leaf plot. Make an observation about the shape, centre or spread of the split stem and leaf plot.

The size of each earthquake that occurred over a three year period, measured from 0 to 9.9, is recorded in a stem and leaf plot. a

What was the size of the largest earthquake measured?

An earthquake of size 6 or greater causes significant damage to buildings. What percentage of the earthquakes caused significant damage to buildings? Round to the nearest whole percent.

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Who is the teacher of the student who can hold their breath the longest?

Explain why the statements are misinterpretations of this stem and leaf plot:

Problem-solving 16, 17a 16

Mathspace WA Curriculum – Year 7 mathspace.co

Stem 0 1 2 3 4 5 6 7 8

Leaf 8 1 3 3 4 5 5 9 2 5 7 7 5 5 8 8 0 1 1 2 2 4 6 6 1 2 6 8 9 0 2 4 4 9 1 8 9 0 4 5 7 Key: 1|2 = 1.2 units

A cyclist measured his heart rate immediately after finishing each event in which he competed. The results are recorded in the stem and leaf plot:

Stem Leaf 16 1 4 4 4 6 8 8 17 0 0 4 6 8 18 0 2 3 3 19 1 2 3 4 7 7 8 Key: 16|1 = 161 beats per minute

What is the range?

Calculate the mean heat rate of the cyclist, to the nearest whole number.

Find the median heart rate.

Find the modal heart rate.

For each extra heartbeat per minute, it takes 7 seconds longer for him to recover. How much longer would it take to recover when he finishes with the fastest heart rate, compared to when he finishes with the slowest heart rate?

This is a partially completed stem and leaf plot showing the ages of 10 people in a community group. Some of the leaves ( A, B and C) are missing. Use the provided mean, median and range to determine the missing values. Mean age: 30

Stem Leaf 2 0 1 2 3 7 B 4 3 4 5 C Key: 2|0 = 20 years old

Median age: 28 Range: 26

16.07 Pie charts After this lesson, you will be able to… • interpret the data represented in a pie chart. • use pie charts to solve problems.

Pie charts Pie chart is a type of graph used to represent data visually, making it easier to understand and interpret. It shows the parts of a whole by dividing a circle into sectors, like slices of a pie. Each sector or “slice,” represents a percentage or proportion of the total. Pie charts are particularly useful for displaying categorical data, especially when you want to compare the relative sizes of the categories. The larger the sector, the larger the percentage of data in that category.

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Fish colours in a tank Orange Red Blue

Looking at this pie chart and using the legend for the categories, we see that the red sector is biggest, so most of the fish are red. The yellow and orange sectors are the same size, so there is an equal amount of yellow and orange fish.

Yellow

The fraction of the circle taken up by each sector indicates what fraction of the total fish are that colour. In this case half the fish are red since the red sector takes up half the circle. We can also write this as a percentage: 50% of the fish are red. Fraction of total

Percentage

Orange

12.5%

Red

50%

Blue

25%

Yellow

12.5%

Notice that the sum of our fractions is 1 and the sum of our percentages is 100%. This is consistent with the fact that a pie chart represents 100% of the data, one whole, split up into sectors representing different categories.

If we know the total number of data points, we can use the percentages represented by the sector sizes to calculate how many data points each sector represents. For example, if there are 80 fish in total, then the number of blue fish is: 25% × 80 = 20 Pie charts may not be the best choice for all types of data. They work best when there are a limited number of categories and when the differences between the categories are clear. If there are too many categories or the percentages are very close, the pie chart may become difficult to read and interpret. In such cases, other types of graphs, like bar graphs or line charts, might be more appropriate.

Example 1

Phys. Ed

Maths

History

Languages

Science

English

ys Ph

a Which was the most popular subject?

Favourite Subject ges Langua .Ed

Every student in year 8 was surveyed on their favourite subject and the results are displayed in the pie chart:

History

English

Maths Science

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Create a strategy

Apply the idea

Check which subject matches the largest sector. Based on the pie chart, English has the largest sector. So the correct answer is option F. b What percentage of the class selected History, Phys. Ed. or Languages? 50%

30%

25%

Create a strategy

Apply the idea

Together these subjects take up one quarter of the circle.

We know that one quarter is 25% in percentage. So the correct answer is option D.

c You later find out that 32 students selected Science. How many students are there in year 8?

Create a strategy

Apply the idea Total Students = 32 x 4

Since the Science sector takes up a quarter of the circle, we can multiply the number of Science students by 4.

= 128

Multiply 32 by 4 Evaluate

Idea summary Pie charts represent 100% of the data, 1 whole, with the different sectors representing different categories. The larger the sector, the larger the percentage of data in that category. The fraction of the circle taken up by each sector indicates what fraction of the total is in that. The sizes can be written as fractions or percentages. To calculate how many data points each sector represents, multiply the percentages represented by each sector by the total number of data points.

16.07 Practice questions Understanding 1

Explorer 1, 2

Adventurer 1, 2

Trailblazer 1, 2

Are these statements true or false? a

The larger the sector, the smaller the percentage of data in that category.

The percentages on a pie chart should always add up to 100%.

Pie charts are used to display categorical data.

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The owner of a business wants to have a company-wide retreat outside of Australia. He narrowed the list down to five countries, but he cannot decide which country to choose. He sends a survey to all 180 of his employees, asking them to choose the country in which the retreat should be held. Would these data displays be appropriate for the owner to use to analyse the results of the survey? a

Dot plot

Fluency 3

Column graph

3–8

Stem and leaf plot

4, 6–9

The pie chart shows the results of a class survey where students were asked to nominate their favourite food: a

Which was the most popular food?

Which two foods were equally popular?

Pie chart

7–9 Favourite Food

Nuggets Pizza Noodles Burger

Ex 1

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The pie chart represents the popularity of each chip brand amongst high schoolers: a

Which brand is the most popular?

Which brand is the least popular?

Which brand is popular with one quarter of the high schoolers?

If 56 students selected Hot Cheezos, how many high schoolers were surveyed?

Edward surveyed a large group of people and asked for their favourite milkshake flavour. The results of the survey are presented in a pie chart. a

What fraction of people chose chocolate?

What percentage of people chose chocolate?

What fraction of people chose strawberry?

What percentage of people chose strawberry?

Mathspace WA Curriculum – Year 7 mathspace.co

Tortillos

Hot Cheezos

Snakis

Wrinkles

Milkshake Flavours

Vanilla

Strawberry

Chocolate

Caramel

A florist recorded the roses of different colours sold in his store over time. This is displayed in the pie chart. If 800 roses were sold in a day, determine the number you would expect to be: a

Red

Orange

White

Pink

Pink 10% Orange 25%

Red 50%

White 15%

The results of a senatorial election in a European country, where 4000 people voted, are displayed in a pie chart. a

Who won the election?

After the election, the winner decides to send promotional packages to everyone they suspect voted for Zorg. How many packages should they prepare?

Voting Results Zorg 25% 10% Umbridge Palpatine 65%

Roald has a weekly income of $1200 and this pie chart shows a break-down of his spending. a

Determine how much money Roald spends weekly on: i

Leisure

Clothes

iii

Food

Housing

If he sticks to his savings plan, how much will he have saved after 5 weeks?

25%

30%

5% 10%

30%

Food

Leisure

Savings

Clothes

Housing

A technology company wishes to see what headphones people are using. They took a small sample and found these: • Apple: 156 • Sony: 19 • Bose: 32 • Other: 74 a

Which sector represents Bose?

What percentage of people use a brand other than Apple, Sony or Bose? Round your answer to one decimal place.

A B

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Reasoning 10

10, 11

Consider the pie chart:

10–12 Fruits purchased from the grocery

If we knew the number of grapes that were purchased, how could we use that information to find the number of apples that were purchased?

Apples

12% 8%

Blueberries Oranges

13%

40%

Grapes

27%

Limes

The pie charts show the distribution of nuclear weapons owned by countries around the world: Nuclear Weapons (Not in USA or Russia) Israel North Korea 20 80

Number of Nuclear Weapons

r Othe 5 116

USA 6450

Pa ki 14 stan 0

Russia 6490

France 300

India 130 China 280

UK 215

Which country has the most nuclear weapons according to the pie charts?

Which two countries have around half of the nuclear weapons that are not owned by Russia or the USA?

How many countries have more than 100 nuclear weapons?

Elise thinks these pie charts are confusing and claims a column graph would be better. Do you agree or disagree with Elise? Explain your reasoning.

This pie chart was displayed on a news program: a Describe two mathematical errors with this pie chart.

Would vote for Grylls

Would vote for McGill

70%

63%

60%

Would vote for Flanders 740

Mathspace WA Curriculum – Year 7 mathspace.co

Which of the images better represents the same information? A

Would vote for Flanders

Would vote for Grylls Would vote for McGill

Would vote for McGill

Would vote for Flanders

After looking at the data, a student concludes that the majority of people would vote for each of the three candidates. Which is not an explanation for how this is possible? A

The three candidates are running for different positions in government.

Three different populations were surveyed.

he survey questions said, “Would you vote for Grylls if they ran for prime minister? T Would you vote for McGill if they ran for prime minister? Would you vote for Flanders if they ran for prime minister?

he survey question said, “If Grylls, McGill and Flanders run for prime minister, who T would you vote for?”

Problem-solving 13, 14 13

Would vote for Grylls

13–15

13–16

A teacher surveys 220 students to find their favourite subject and wishes to construct a pie chart to display the data. The survey results are recorded in the table: Subject

Number of students

Maths

English

History

Science

Other

Percentage

Angle size 20% of 360 = 72°

Use your calculator to complete the table and find the percentage of students and angle size for each subject.

Construct a pie chart to represent this data, labelling each sector piece with the name and percentage.

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A pharmaceutical company runs trials of a potential virus vaccine. There are 1080 volunteers for the trial and the results are recorded in the table: Result

Improvement

705

No change

240

Deterioration

135

Use your calculator to complete the table and find the angle size for each result.

Construct a pie chart to represent this data, labelling each sector piece with the name and angle size.

A business wishes to construct a pie chart to display which month holidays are taken over the summer months. They surveyed 180 employees and the results are recorded in the table: Number of employees

November

December

January

February

March

Angle size

Use your calculator to complete the table and find the angle size for each month.

Construct a pie chart to represent this data, labelling each sector piece with the name and angle size.

A marketing company wishes to construct a pie chart to display the market share of smartphone manufacturers. They surveyed 4000 people and the results are recorded in the table: Manufacturer

742

Angle size

Month

Number of volunteers

Responses

Apple

2051

Samsung

967

Huawei

531

Other

451

Angle size

Use your calculator to complete the table and find the angle size for each manufacturer, rounded to one decimal place.

Construct a pie chart to represent this data.

Mathspace WA Curriculum – Year 7 mathspace.co

16.08 Line graphs After this lesson, you will be able to… • interpret the data represented in a line graph. • use line graphs to solve problems.

Line graphs A line graph is a type of chart displaying points connected by straight line segments. This is used to show how one value varies as another value changes. Line graphs are usually used for continuous data to show trends in data over time. For example, a line graph may show temperature in a city over a day. The graph would help us easily find the temperature at a particular time of day, the maximum or minimum temperature and other trends such as at what times the temperature was increasing or decreasing. When looking at a line graph it is essential to read and understand the information given. Important features of line graphs include: • a clear title • labels on both axes, with units where appropriate • a clear scale on each axis Sometimes other important details about the graph are included. Reading and understanding this information helps you understand the graph better and make sense of what it is showing.

Example 1 The line graph shows the number of ice creams sold over the course of a day: Ice Cream Sales

Number of Ice Creams Sold

30 25 20 15 10 5 10 p.m. 11 p.m. 12 p.m. 1 p.m. 2 p.m. 3 p.m. 4 p.m. 5 p.m. 6 p.m. Time

a At what time of day were the least amount of ice creams sold? A

6 pm

4 pm

12 pm

10 pm

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Create a strategy

Apply the idea

The vertical axis tells us how many ice creams were sold at a certain time. The least number of ice creams will be shown when the line graph reaches its lowest height.

The lowest height where only 5 ice creams were sold happened at 4 pm. The correct option is B.

b What were the most amount of ice creams sold at a particular time?

Create a strategy

Apply the idea

The vertical axis tells us how many ice creams were sold at a certain time. Look at the time where the line graph is at its highest. This will show the most amount of ice creams sold.

The line graph is highest at 1 pm. The point on the vertical axis that matches this time is 26, so the most amount of ice cream sold is 26.

c There are two peak hours for ice cream sales, at lunch time (1 pm) and in the evening (6 pm). What was the difference in sales between these two peaks?

Create a strategy

Apply the idea

We found in part (b) that 26 ice creams were sold at 1 pm. We want to find the difference between this and the amount of ice creams sold at 6 pm. What operation will we have to use to find this difference?

Difference in sales = 26 − 24 Subtract 24 from 26 =2

Evaluate

Idea summary A line graph is a type of chart displaying points connected by straight line segments. Important features of line graphs include: • a clear title • labels on both axes, with units where appropriate • a clear scale on each axis

16.08 Practice questions Understanding

Explorer 1, 3

Adventurer 1–3

Trailblazer 1–3

What do line graphs tell about the data being presented?

How might a scale on a line graph that is too big or too small impact our interpretation of the data presented?

744

Mathspace WA Curriculum – Year 7 mathspace.co

Match each scenario to the most suitable type of data display: i

Dot plot

A statistician surveyed 100 people to find out which eye colour is most common.

An athlete wants to identify patterns in their heartbeat during their workout.

A teacher asks her class of 25 students to tell her their favourite subject in school to see how many students prefer maths over science.

Fluency 4

Pie chart

iii

4–6, 8, 9, 10ab, 15a-d, 16, 17

Line graph

7, 8, 10–12, 15, 16

The line graph shows the temperature over a five-day period in a particular city. Determine the city’s temperature on these days: a

Day 1

Day 3

Day 5

Day 2

Temperature in ºC

7, 10–16

40 39 38 37 36 35 34 33 32 31 30 1

The line graph represents the number of visitors to a museum over a 12-month period: a

How many visitors were there in Month 1?

How many visitors were there in Month 10?

In which month(s) did the museum have 2550 visitors?

Were there more visitors in Month 7 than in Month 9?

Number of visitors

3 Day

2850 2800 2750 2700 2650 2600 2550 2500 2450 2400 2350 2300 0

1 2 3 4 5 6 7 8 9 10 11 12 Month

The table shows the number of different vegetables Susana ate on each day of the week. Construct a line graph for this information. Day No. of Vegetables

Sun 4

Mon 6

Tue 1

Wed 1

Thu 3

Fri 2

Sat 5

Dylan writes a new song and asks his friends to give it a score from 0 to 1. Construct a line graph for this information. Score Frequency

0.0 8

0.1 3

0.2 8

0.3 5

0.4 6

0.5 9

0.6 10

0.7 9

0.8 9

0.9 8

16.08 Line graphs mathspace.co

1.0 9

745

What year had the lowest seal population?

One year, due to too much fishing in the local area, a shortage of fish caused the seal population to drop. In which year did the seal population drop?

Number of seals

Seal population at Rocky point 55 50 45 40 35 30 25 20

2013

2014

2015

What is the highest seal population over the five years?

By how much did the seal population increase over the five years?

The highest daily temperature in Sommersville over a week is presented in the line graph: a

Which day had the warmest temperature?

Which two days had the same temperature?

How much warmer was Friday than Monday?

2016

2017

Year

Daily Temperature 26º 24º 22º 20º 18º 16º 14º

Mon

Tue

Wed

Thu

Fri

Sat

Sun

Day

A paper corporation records the amount of paper they produce each month over a year in a line graph. a

746

In which month did the paper corporation produce the most paper?

Paper Produced 4.0 3.8

Mass (tonnes)

Ex 1

The line graph shows the seal population at Rocky Point over a 5-year period:

Temperature ( Degrees Celsius)

3.6 3.4

In which month did the paper corporation produce the least paper?

In which period did paper production increase the most?

In which period did paper production decrease the most?

In which period did paper production not change?

Mathspace WA Curriculum – Year 7 mathspace.co

3.2 3.0

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

Month

Sarah records her speed in metres per second at various points on a 1.5 km running track on a line graph. 5

Speed (m/s)

4.8 4.6 4.4 4.2 0.3 km

0.6 km

0.9 km

1.2 km

1.5 km

Distance

At which three distances did Sarah reach a peak speed?

At which two distances did Sarah reach a trough speed (low point)?

How far did Sarah travel between the second trough and the third peak?

By how much did Sarah’s speed increase between the second trough and the third peak?

The line graph shows the amount of petrol (in litres) in a car’s tank during a long drive. Given that the drive started at 8 a.m.: How much petrol was initially in the tank?

What happened at 9 a.m. and 1 p.m.?

Calculate the petrol used between 1 p.m. and 5 p.m.

To the nearest hour, state the time the petrol in the tank first decreased to the 18 L mark.

The line graph shows the median house prices per quarter for the four years to September 2013.

4 5 Hours

$600k

Sep -12 Dec -12 Mar13 Jun-1 3 Sep -13

$400k

$500k

Jun-

Determine the range of prices over the four year period to the nearest $100 000. List the quarters in which the three highest median house prices occurred.

Mar-

Describe the underlying trend in median house prices over the four years.

Se p-0 De 9 c-0 9 Ma r-10 Jun -10 Se p-1 0 De c-10 Ma r-11 Jun -11 Sep -11 Dec -11

30 28 26 24 22 20 18 16 14 12 10

$700k

Median house price

Petrol (litres)

16.08 Line graphs mathspace.co

747

The line graph shows monthly rainfall data for Berlin and London. a b

748

65 60

Calculate the range in monthly rainfall for Berlin between June and October. In the months between March and July (inclusive), determine the range in London’s monthly rainfall.

55 50 45 40 35 30

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

London

Berlin

Which city has the lowest range over the year?

In which month did the cities have the smallest difference in rainfall?

The price of cans of soft drink over a year is represented on a line graph. a

What was the lowest price of the cans of soft drink?

In which months was the price the lowest?

What was the highest price of the cans of soft drink?

Soft Drink Prices $3.00 $2.75 $2.50 $2.25 $2.00 $1.75 $1.50 $1.25 $1.00

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

Month

In which months was the price the highest?

If 250 cans of soft drink are sold each month, calculate the difference in the value of the sales between the month with the highest price and the month with the lowest price.

Luigi brings a 1.8 L bottle of water to work each day. He records the amount of water in the bottle at the start of each hour on a line graph. a

How much water was in Luigi’s bottle at the start of the day?

What time was Luigi’s bottle first refilled?

How much water did Luigi drink in the hour after first refilling his bottle?

Water Bottle

Water (litres)

Monthly Rainfall (mm) in Major Cities 70

Which city recorded the highest monthly rainfall over the year?

Price

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

9 am 10 am 11 am 12 pm 1 pm 2 pm 3 pm 4 pm 5 pm Time

How many hours were between the first and second time Luigi filled his bottle?

How much water did Luigi drink between the two refills?

How much did the amount of water in Luigi’s bottle change between the start and end of the day?

How much water did Luigi drink in total during the day?

Mathspace WA Curriculum – Year 7 mathspace.co

Danielle’s distance from home throughout the day is recorded on a line graph. a b c d

How far was Danielle from her house at 2:00 p.m.? What time did Danielle get back home?

3.5 3.0

What was the furthest distance Danielle was from her house? Determine the total time that Danielle was away from her house.

Reasoning 18

4.0

Distance (km)

18, 19, 22

2.5 2.0 1.5 1.0 0.5 0.0

11:00 a.m.

11:30 a.m.

12:00 12:30 p.m. p.m.

1:00 p.m.

1:30 p.m.

2:00 p.m.

2:30 p.m.

3:00 p.m.

Time

18, 19, 21

19–21

The manufacturer of a dishwasher wants to create a graph showing how one of their dishwasher models work. These are the stages of its operation: 1.

The dishwasher quickly fills to half-way and performs a short wash.

2. The dishwasher empties completely. 3. The dishwasher completely fills up quickly and remains full over the course of the wash. 4. The dishwasher empties completely. Which graph matches the water level inside the dishwasher throughout a cycle? A

y Full

Full

Empty 10 20

Time (mins) 40 50

y Full

Empty 10 20

Time (mins) 40 50

y Full

Time (mins) 40 50

Empty 10 20

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A kitchen sink starts with 20 L of water in it. It empties at a rate of 4 L per minute and then after 4 minutes the drain gets blocked. No water empties for 2 more minutes while it gets unblocked, then the remaining water is drained out in 1 minute. Construct a line graph that represents this scenario.

A tank is being filled at a constant rate of 4 L per second for 4 minutes. The hose is then turned off for 2 minutes. Finally it is topped up again with water at a rate of 3 L per second for 1 minute. Construct a line graph that represents this scenario.

The travel graph shows the first ten minutes of a student’s journey home from school. 10 9 Distance (km)

8 7 6 5 4 3 2 1 0

2 3 4 5 6 7 8 9 10 Time (mins)

Describe their travel between these times:

0 to 3 minutes

3 to 6 minutes

6 to 9 minutes

9 to 10 minutes

Consider the line graph of the sales of a car company:

Cars sold

Tiger car sales 100 90 80 70 60 50 40 30 20 10 0

3 Month

Explain why the statements are misinterpretations of the line graph.

750

The company achieved the target of increasing trend in sales for the year.

The changes in the sales each month are almost consistent.

Mathspace WA Curriculum – Year 7 mathspace.co

23–25

Two siblings, Apollo and Artemis, leave home at different times. They are travelling to see their favourite sports team compete in the championship. The stadium is 30 km away and the match starts at 9:00 p.m. a

At what times did they both leave home?

Between which two times were Apollo and Artemis the same distance from home?

Who was travelling faster?

Did Apollo make it to the stadium on time?

24–27

30 Distance (km)

Problem-solving 24, 25

25 20 Artemis

15 Apollo

10 5

2 3 4

5 6 7

8 9 10

Time (p.m.)

This travel graph represents a train’s journey between the airport and a nearby station. a

How far did the train travel over the 10 minute period?

When was the train travelling the fastest?

Distance (km)

10 9 8 7 6 5 4 3 2 1 0

Kenneth works for a delivery company. The graph shows his distance from the warehouse throughout his shift. a

How far was Kenneth from the warehouse at the start?

How far was he from the warehouse after 8 hours?

How many times did he visit the warehouse during this time period?

Calculate the total distance that he travelled between 1 and 4 hours.

When was Kenneth travelling the fastest?

9 8 Distance (km)

1 2 3 4 5 6 7 8 9 10 Time (mins)

7 6 5 4 3 2 1 0

4 5 6 7 Time (hours)

8 9

16.08 Line graphs mathspace.co

751

The Weber family travel 600 km every year for their annual holidays. Their distance from home on the trip this year is recorded in a travel graph. a

When did they stop for a break?

How far from their destination were they after 2 hours?

600 Distance (km)

Identify two time periods when they were travelling at the same speed.

500 400 300 200 100

Castor and Pollux are working on a bushfire prevention team. Over the course of their 10-hour shift they move along a boundary and remove any flammable material they find. The travel graph shows their distance from the base throughout their shift: a

How far from the base was Pollux when the shift began?

How far apart were Castor and Pollux after 2 hours?

When did Castor and Pollux meet up?

When was Pollux travelling the same speed as Castor’s speed in his first three hours?

Distance (km)

10 9 8 7 6 5 4 3 2 1 0

3 4 5 6 7 Time (hours)

Pollux Castor 1

2 3 4 5 6 7 8 9 10 Time (hours)

Investigation: Statistical investigations

Investigate online

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Mathspace WA Curriculum – Year 7 mathspace.co

mathspace.co

16 Chapter review 1

Identify the type of data described if discrete, continuous, nominal or ordinal: a

Number of students in each classroom in a school

Names of different colors in a box of crayons

Temperatures recorded each day for a month

Type of data for the ratings (1 – 5 stars) given to a restaurant

A student scored these grades on his end of year tests: 92, 91, 88, 89, 93, 86, 90, 92, 96 a

How many tests did the student take?

Calculate: i

Mean

Median

iii

Mode

Examine the data set presented in the frequency table. Suppose one score of 7 is changed to a 10. Determine if each measure would change or remain the same:

Mean

Median

Mode

Range

Score 2 3 5 6 7 9 10

Range Frequency 1 2 2 1 3 2 1

Olivia has a fitness goal and records her exercise routine. She used a dot plot to track the number of days she exercised each week for a certain period. a

In this dot plot, what does each dot represent?

In how many weeks did she exercise for 5 days?

What was the most frequent number of days she exercised per week?

1 2 3 4 5 6 7 Days per week spent exercising

Chapter 16 review mathspace.co

753

Consider the column graph showing the eye colours of the students in a class: green, blue, hazel and brown.

Determine whether these statements are true or false: a

Hazel eyes are the least common among the students.

Two children have green eyes.

Brown eyes are the second most common eye colour.

d 6

Number of students

8 6 4 2 0

Green

No child has brown eyes.

Blue Hazel Brown Eye colour

A student conducts a survey observing the colours of vehicles passing through an intersection. The results are displayed in the column graph.

Colours of vehicles passing through intersection 12

Frequency

10 8 6 4 2 0 Black

White

Blue

Green

Silver

Yellow

Red

Vehicle colour

Which is the most popular vehicle colour?

Which two colours were equally popular?

How many vehicles were observed?

The stem and leaf plot shows the number of books read per year by a group of people. Identify the outlier.

Sam is using a special tool to measure the height of trees (in metres) in a specific area of a forest. He records his findings in the stem and leaf plot. a

Find the median tree height.

Find the range of tree heights.

754

Number of books Leaf 7 8 8 9 9 0 1 1 2 2 3 4 4 5 9 Key: 1|7 = 17 books

Stem 1 2 3

Stem 0 1 2 3

3 0 2 0

Leaf 4 5 8 9 0 3 5 6 7 7 7 3 3 4 5 6 6 7 9 8 0

Key: 1|0 = 10 metres Sam finds that the sum of all the tree heights is 568 m. Find the mean tree height, rounded to two decimal places.

Mathspace WA Curriculum – Year 7 mathspace.co

Fish colours in a tank

This pie chart represents the distribution of fish colours in a tank. a

Which colour is the most common among the fish?

Which fish colour makes up one quarter of the fish in the tank?

If there are 28 blue fish, how many fish are there in total in the tank?

This pie chart represents the distribution of Year 10 students participating in various sports. a

Which sport has the most participants according to the pie chart?

Which sport has the least participants according to the pie chart?

How many students participate in football and cricket combined?

The number of books sold by a bookstore over a 12-month period is represented in this line graph. a

How many books were sold in Month 1?

How many books were sold in Month 10?

In which month(s) did the bookstore sell 2100 books?

Were more books sold in Month 7 than in Month 9?

Orange

Blue

Red

Yellow

Year 10 sports by student Basketball

Cricket

30 86 53

Volleyball 27

2350 2300 2250 2200 2150 2100 2050 2000 1950 1900 1850 1800

Water Polo

Football

Number of books

Month 1 2 3 4 5 6 7 8 9 10 11 12

A financial analyst is studying the growth of monthly sales in a retail store. They collected data on the percentage increase in sales for 20 consecutive months. The results were: 2, 2, 4, 4, 5, 5, 7, 7, 8, 8, 9, 9, 10, 13, 13, 14, 16, 17, 20, 46 The analyst wants to highlight the significant growth potential of the retail store. Which measure of central tendency should they use in their report? Explain your reasoning.

Chapter 16 review mathspace.co

755

Consider the stem and leaf plot:

Stem 2 3 4 5 6

Which measure of centre best represents the centre of the data? Explain your reasoning.

Leaf 6 7 8 9 2 3 3 4 5 2 3 4

Key: 1|2 = 12 14

A group of employees in a company are tested for their typing speed. The participants are given a 5-minute typing test and their typing speeds, in words per minute, are listed: 60, 70, 65, 60, 68, 55, 57, 59, 59, 61, 100, 59, 61, 61

Identify any outliers.

Which is the most likely explanation for the outlier? A

The employee is a professional typist.

The employee used typing software to assist them.

The employee practised more than others.

The timer malfunctioned.

A library tracks the number of books its patrons borrow each week, rounded to the nearest whole number. The dot plot represents the data for the past month. a

Find the median number of books borrowed by patrons each week.

What fraction of patrons borrowed more books than the median amount?

If a patron borrows more than 1 book per week, they are entitled to a $5 gift card. How much money did the library spend on gift cards over the last month?

Number of Books Patrons Borrow Each Week

A school conducted a survey to determine the time (in minutes) students spend on homework each day. The survey results are: 35, 42, 25, 38, 30, 27, 33, 45, 29, 40, 36, 28, 32, 31, 34 Represent this data as an ordered stem and leaf plot and comment on the distribution.

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Mathspace WA Curriculum – Year 7 mathspace.co

Two friends, John and Jane, leave home at different times. They are going for a bike ride to a nearby park that is 20 km away and they plan to meet there at 5:00 p.m. Distance 30 km 25 km Jane

20 km 15 km 10 km

John

5 km Time (p.m.) 2 3 4 5 6 7 8 9 10

At what times did they both leave home?

Between which two times were John and Jane the same distance from home?

Who was travelling faster?

Did John make it to the park on time?

Mia and Ava are participating in a beach cleanup event. Over the course of their 8-hour shift they move along the beach and collect any rubbish they find. This travel graph shows their distance from the starting point throughout their shift: 10

Distance (km)

9 8 7 6 5 4 3

Ava

Mia

Time (hours) 1

How far from the starting point was Ava when the shift began?

How far apart were Mia and Ava after 2 hours?

When did Mia and Ava meet up?

At what point during her shift was Ava travelling at the same speed as Mia?

Trent was working with a set of 6 data points. He knows 5 of them are: 11, 12, 17, 25 and 28. a

If he knows the mean was 19, what is the missing score?

If it was actually the median that was 19, what is the missing score?

Add two more scores to the data set that would also keep the mean and the median value the same. Chapter 16 review mathspace.co

757

Big ideas Understanding time units, time representations and time zone conversions can help efficiently measure and calculate time differences.

Curriculum Outcomes Explore and interpret representations of time zones within Australia using 12and 24-hour time and determine the local time at different locations considering different times of the year

17 Time Chapter outline 17.01 Elapsed time 17.02 12 and 24-hour time 17.03 Australian time zones Chapter 17 review

760 765 769 775

Pilots and aircrew have to adjust to different time zones during flights. A flight from Sydney to Los Angeles crosses over 17 time zones!

17.01

Elapsed time

After this lesson, you will be able to… • convert between seconds, minutes and hours. • calculate time differences.

Time conversions We use these conversions to change between different units of time: • 1 minute = 60 seconds • 1 hour = 60 minutes • 1 day = 24 hours Remember there are 7 days a week and 52 weeks a year. A year, deﬁned by the time for Earth to orbit the Sun, is approximately

days. Every year has 365 days, except every fourth year,

called a leap year, which has 366 days to account for the extra quarter day. Leap years have an additional day in February. As we do for other types of conversions, we use multiplication and division to change between time units as follows: ÷60

÷60

Seconds

Minutes

×60

÷24

Hours

×60

Days

×24

For example, if we want to know how many seconds are in 5 minutes of time, we can multiply the number of minutes by 60 because there are 60 seconds in each minute. 5 minutes × 60 = 300 seconds So, there are 300 seconds in 5 minutes.

Example 1 It takes around 300 minutes to drive from Sydney to Thredbo. How many hours is this?

Create a strategy Divide the number of minutes by the number of minutes in an hour.

760

Mathspace WA Curriculum – Year 7 mathspace.co

Apply the idea 1 hour = 60 minutes, so we need to divide by 60. Write as a fraction

Remove the zeros

Divide 30 by 6

The drive takes 5 hours.

Example 2 Han boils an egg for 360 seconds. How many minutes is this?

Create a strategy

Apply the idea

Divide the number of seconds by the number of seconds in a minute.

1 minute = 60 seconds, so we need to divide by 60. Write as a fraction

Divide 360 by 60

Han boils an egg in 6 minutes.

Idea summary Conversions between the different units of time: ÷60

Seconds

÷60

Minutes

×60

÷24

Hours

×60

Days

×24

Elapsed time Now let’s explore how to calculate changes in time using a clock.

Interactive exploration

Discover this concept in action online

mathspace.co

17.01 Elapsed time mathspace.co

761

When adding or subtracting times, we need to be aware that there are 60 minutes in an hour. If the minutes add to 60 or more, increase the number of hours. The time will change between a.m. and p.m. whenever the hour changes from 11 to 12.

Example 3 Which of these represents 30 minutes after 11:40 p.m.? A 12:10 a.m.

11:10 p.m.

11:70 p.m.

12:10 p.m.

Create a strategy

Apply the idea

When adding times, if the minutes add to 60 or more, increase the number of hours. The time will change between p.m. to a.m. whenever the hour changes from 11 to 12.

Add the 30 minutes by adding 20 and then 10 minutes. 11 : 40 p.m. + 20 minutes = 12 : 00 a.m. Add the remaining 10 minutes: 12 : 00 a.m. + 10 minutes = 12 : 10 a.m. 30 minutes after 11:40 p.m. will be 12:10 a.m., option A.

Example 4 Kate leaves her home at 11:15 a.m. to visit her grandmother in a different state. She arrives at her grandmother’s house at 5:15 p.m. How long was Kate’s trip in hours?

Create a strategy

Apply the idea

Since both times have the same number The number of hours from 11:00 a.m. to of minutes, we can focus on the difference 12:00 midday is 1. The number of hours between between the hours. Find the difference between 12:00 midday and 5:00 p.m. is 5. 11:00 a.m. and 5:00 p.m. Length of trip = 1 + 5 Add the hours = 6 hours

Evaluate

Idea summary Remember that when adding times, if the minutes add to 60 or more, increase the number of hours. The time will change between a.m. to p.m. whenever the hour changes from 11 to 12.

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17.01 Practice questions Understanding 1–2 1

⬚ seconds = 1 minute

Trailblazer 2–3

60 minutes = ⬚ hour

⬚ hours = 1 day

⬚ days = 1 week

A number of hours to minutes

A number of seconds to minutes

A number of hours to days

A number of minutes to hours

A number of hours to seconds

A number of days to minutes

Is the time after these changes written with a.m. or p.m.? a

6:30 p.m. + 12 hours

1:00 a.m. − 2 hours

9:05 a.m. − 9 hours and 5 minutes

4:20 p.m. + 24 hours and 20 minutes

Fluency 4

Describe how to convert these units of time. a

Adventurer 2–3

Complete the statements: a

Explorer

4a–e, 5–8, 10aceg, 12, 14, 16

4–8, 10aceg, 13, 14, 16

4, 7–9, 10bdfh, 11, 14, 15

Convert the times: a

9 days to hours

11 minutes to seconds

36 hours to minutes

24 hours to seconds

240 seconds to minutes

360 minutes to hours

780 seconds to minutes

780 minutes to hours

Ex 1

It takes 150 minutes from Melbourne to Queensland by air. How many hours (and minutes) is this?

Ex 2

Jean can run a lap for 180 seconds. How many minutes is this?

Neil’s swimming lesson went from 11:15 a.m. until midday. How many minutes was Neil’s lesson?

A YouTube video started playing at 1:40 p.m. and ﬁnished at 2:18 p.m. How long is the video in minutes?

How many hours are there in a non-leap year?

Find the new time:

Ex 3

30 minutes after 9:30 a.m.

40 minutes after 11:40 p.m.

9:30 a.m. plus half an hour

7:00 p.m. plus 24 minutes

45 minutes after 11:20 a.m.

20 minutes after 3:50 p.m.

10:30 a.m. plus two and a half hours.

9:10 p.m. plus 63 minutes

A song clip starts playing at 7:55 p.m. and finishes at 8:11 p.m. How long is the song clip in minutes? 17.01 Elapsed time mathspace.co

763

Ex 4

Sally’s cruise ticket says that it begins at 10:00 a.m. and ends at 2:00 p.m. How long is the cruise in hours?

James leaves home at 4:00 p.m. and gets home at 8:00 p.m. How many hours was James out?

Jake leaves his home at 8:10 a.m. to visit his aunt in a different state. He arrives at his aunt’s house at 4:10 p.m. How long was Jake’s trip in hours?

Eileen starts watching a movie at 11:15 a.m., which goes for 3 hours and 45 minutes. What time will the movie ﬁnish?

A computer starts updating at 1:00 p.m. The update will go for 130 minutes. Find the time that the update will be ﬁnished.

Reasoning 17

17, 18

18, 19

Josie is trying to calculate the time 42 minutes after 9:45 a.m. Opie tells her to just add 42 and 45. Josie does this and gives her answer as 9:87 a.m. a

Explain why this answer cannot be correct.

What would you add to Opie’s instructions to help Josie get the correct answer.

We usually use 52 weeks to represent a year. Is this the closest number of weeks for both a normal year and a leap year? Explain.

Complete the calculation: a

Add 2 hours and 20 minutes to 1 hour and 15 minutes.

Add 2 hours and 20 minutes to 1 hour and 40 minutes.

Add 2 hours and 30 minutes to 1 hour and 40 minutes.

For each of the conditions for the sum of b and d minutes given below add a hours and b minutes to c hours and d minutes given a, b, c and d are integers. d

b + d < 60

b + d = 60

b + d > 60

Problem-solving 20–22 20

20–23

22–25

Today is Dave’s birthday and he just turned 9 years old. Assume there are 365 days in a year. Find his age in: a

Days

Hours

Minutes

Seconds

The average person sleeps for 8 hours a night. How many weeks does the average person spend asleep over 5 years? Assume there are 365.25 days in a year. Give your answer to the nearest week.

If an average person showers for 6 minutes per day, ﬁnd how many hours they spend showering in: a

764

One year (assume non-leap year)

Mathspace WA Curriculum – Year 7 mathspace.co

55 years (ignoring leap-years)

A year on Mars is 687 Earth days. Xavier lives in a Martian settlement and has just celebrated his 5th birthday, measured in Martian years. Assuming each Earth year is 365 days, how old would Xavier be in Earth years, in years and months, to the nearest month.

A day on Mars is 24 Earth hours and 37 Earth minutes. A year on Mars is 687 Earth days. How long is a Martian year in Martian days? Give your answer to the nearest Martian day.

Augustus is working on a scheme to decimalise the day. He wants each day to be divided into 10 decimal hours, each decimal hour to be divided into 100 decimal minutes and each decimal minute to be divided into 100 decimal seconds. a

Find the length of a decimal hour in minutes.

Find the length of a decimal minute in seconds.

Find the length of a decimal second in seconds.

17.02

12 and 24-hour time

After this lesson, you will be able to… • convert between 12 and 24-hour times.

12 and 24-hour time There are two ways we represent time over a day: 12-hour and 24-hour time. The following diagram shows how 12 and 24-hour time are related: 24 hour time 00:00

04:00

08:00

12:00

16:00

20:00

00:00

12:00 am

4:00 am

8:00 am

12:00 pm

4:00 pm

8:00 pm

12:00 am

12 hour time

12-hour time represents a day as two blocks of 12 hours using “a.m.” and “p.m.” to represent before and after midday respectively. 24-hour time represents time as one block of 24 hours. There are no “a.m.” or “p.m.” indicators in this format. Instead, the hours are numbered from 00 (midnight) to 23 (11 p.m.). For example: • 8:45 a.m. is 08:45 • 3:15 p.m. is 15:45

Interactive exploration

Discover this concept in action online

mathspace.co

17.02 12 and 24-hour time mathspace.co

765

To convert between 12 and 24-hour times: • In 24-hour time 12:00 a.m. and 12:00 p.m. are 00:00 and 12:00 respectively • Other a.m. times are the same in 24-hour format with the addition of a zero before the digit representing the hour • Other p.m. hours are converted by adding twelve to the hours digit to get 24-hour time • These processes are reversed to convert between 24 and 12-hour times

Example 1 Convert 3:41 p.m. to 24-hour time.

Create a strategy Add 12 to the given hour.

Apply the idea 3:41 p.m. to 24-hour time = 3 : 41 + 12 : 00 = 15 : 41

Add 3 and 12 Evaluate

So, the 24-hour time of 3:41 p.m. is 15:41.

Example 2 Convert 14:54 to 12-hour time.

Create a strategy Subtract 12 from the given hour and add “p.m.” at the end.

Apply the idea 14:54 to 12-hour time = 14 : 54 − 12 : 00

Subtract 12 from 14

= 2 : 54 p.m.

Evaluate and add p.m. at the end

So the 12-hour time of 14:54 is 2:54 p.m.

Idea summary Conversion between 12 and 24-time: 24 hour time 00:00

04:00

08:00

12:00

16:00

20:00

00:00

12:00 am

4:00 am

8:00 am

12:00 pm

4:00 pm

8:00 pm

12:00 am

12 hour time

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Mathspace WA Curriculum – Year 7 mathspace.co

17.02 Practice questions Understanding 1–3

Describe how to convert these time formats:

A 12-hour time (p.m.) to 24-hour time

A 24-hour time (after midday) to 12-hour time

How does the 12-hour time format differ from the 24-hour time format?

4cdgh, 5b–d, 6ac, 7, 8, 10, 11, 15, 17, 18

10:30 a.m.

9:15 p.m.

4:12 a.m.

8:40 a.m.

7:25 p.m.

4:31 a.m.

12:47 a.m.

6:13 p.m.

00:45

13:59

00:56

Convert these times to 12-hour time format: 05:30

18:30

Convert these 24-hour clock times to a.m. or p.m. times: 09:48

12:28

22:47

Find the time and day that is: a

19 hours and 20 minutes after 20:15 on Thursday

17 hours and 15 minutes before 15:25 on Thursday

22 hours and 58 minutes after 19:25 on Monday

18 hours and 39 minutes before 14:12 on Monday

Saturday 13:15 − 15 hours 18 minutes

Thursday 13:20 + 21 hours 59 minutes

Express these times in 24-hour time: a

Quarter to midnight

Twenty past one in the afternoon

Half past ten at night

Quarter past eleven in the morning

Ten past eight in the morning

Ten to four in the afternoon

A clock shows the time 19:39. If it is known that the clock is running slow by 31 minutes, what is the actual time: a

4abef, 5ab, 6acd, 7a–d, 8, 10–12, 14, 16, 18

a 7

4abef, 5ab, 6, 7ab, 8, 9, 11, 13, 14, 16, 17

Convert these times to 24-hour time format :

a 6

Trailblazer 1–3

Why do we use “a.m.” and “p.m.” in the 12-hour clock system?

Fluency

Ex 2

Adventurer 1–3

Ex 1

Explorer

in 24-hour time

in 12-hour time

A clock shows the time 17:27. If it is known that the clock is running fast by 40 minutes, what is the actual time: a

in 24-hour time?

in 12-hour time?

17.02 12 and 24-hour time mathspace.co

767

Write the time shown on each of the clocks in 24-hour time: a

A taxi departs at 13:32 from Sydney and arrives at 20:14 at Melbourne. How long did the taxi take for the trip?

David and his family visited his grandfather for his birthday lunch. They left at 2:50 p.m. What is this time in 24-hour time?

Amy baked cookies for an afternoon tea party for her friends. The cookies take 35 minutes to cook. Amy puts them in the oven at 3:05 p.m. What time did she take the cookies out of the oven? Express your answer in 24-hour time.

Mary’s mum said that she would meet her at the mall for coffee at 5:20 p.m. What is this time in 24-hour time?

Gabby had friends over for a dinner party that went on for 3 hours. If they started eating at 18:20, at what time did they ﬁnish? Express your answer in 24-hour time.

A train ride between two cities takes 6 hours and 49 minutes. If you leave one city at 21:40, at what time will you arrive at the other city? Express your answer in 24-hour time.

A ship arrives at 17:28 in Melbourne. If the trip took 2 hours and 16 minutes, at what time did the ship depart from Sydney? Express your answer in 24-hour time.

Reasoning 19

19, 20

Kirk says that it is easy to convert from 12-hour times to 24-hour time, all you need to do is add 12 to the hours. a

Explain why Kirk is incorrect.

Give one example where his method would work and one where it would not.

Make a correction to Kirk’s statement so that it will always work.

Put these times in order from earliest in the day to latest in the day: 12:25 p.m., 00:30, 19:30, 12:25 a.m., 8:25 p.m., 13:25

Problem-solving 21, 23

22, 23

22–24

A shop opens at 10:00 a.m. and closes at 6:30 p.m. If it takes the shop owner 30 minutes to set up and 20 minutes to close down, what is the total time the owner spends at the shop?

Indi is taking a ﬂight from Sydney to Tokyo which departs at 9:40 a.m. local time and takes 9 hours and 50 minutes. If she had to be at the airport in Sydney 3 hours before her ﬂight and it takes her one and a half hours to collect luggage and clear immigration in Tokyo, how long is her journey from arriving at the airport in Sydney to leaving the airport in Tokyo?

768

Mathspace WA Curriculum – Year 7 mathspace.co

A digital clock shows 22:50 and is losing 2 minutes every hour.

What time will it show in 6 hours?

What time did it show 2 hours ago?

A clock is showing 8:50 p.m. and is gaining 3 minutes every hour. a

What time will it show in 7 hours?

What time did it show 4 hours ago?

17.03

Australian time zones

After this lesson, you will be able to… • the time difference between Australian states and territories using time zones..

Time zones in Australia Australia is divided into three time zones as shown in the image: • Australian Eastern Standard Time (AEST) • Australian Central Standard Time (ACST) • Australian Western Standard Time (AWST) AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

2h behind

0.5 h behind

Compared to AEST

NT QLD WA SA NSW VIC TAS

17.03 Australian time zones mathspace.co

769

Australia also uses Daylight Saving Time (ADST) in the summer months. From October to April, some of the Australian states and territories turn their clocks forward one hour. There are two different ADST zones as shown in the map: AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

ACDT (UTC + 10.5)

AEDT (UTC + 11)

2 h behind

0.5 h behind

Compared to AEST

0.5 h ahead

1 h ahead

NT QLD

QLD

WA SA

SA NSW

NSW

VIC

TAS NO DST

TAS DST

Example 1 When the time is 14:15 in Perth, what is the time in: a Sydney during non ADST?

Create a strategy Refer to the left side of the image above.

Apply the idea Sydney (in NSW) is UTC + 10 while Perth (in WA) is UTC + 8. This means that Sydney is 2 hours ahead of Perth. Sydney time = 14 : 15 + 2 hours = 16 : 15

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Mathspace WA Curriculum – Year 7 mathspace.co

Add the 2 hours Evaluate

b Sydney during ADST?

Create a strategy Refer to the right side of the image above.

Apply the idea Sydney (in NSW) is now in AEDT which is UTC + 11 while Perth (in WA) is still in UTC + 8. This means that Sydney is 3 hours ahead of Perth. Sydney time = 14 : 15 + 3 hours

Add the 3 hours

= 17 : 15

Evaluate

Idea summary Australia is divided into three time zones: • Australian Eastern Standard Time (AEST) UTC + 10 • Australian Central Standard Time (ACST) UTC + 9.5 • Australian Western Standard Time (AWST) UTC + 8 Some states in Australia observe two Daylight Saving Time (ADST) zones from October to April: • Australian Central Daylight Time (ACDT) UTC + 10.5 • Australian Eastern Daylight Time (AEDT) UTC + 11

17.03 Practice questions Understanding 1–3

Explorer

Adventurer

Trailblazer 2, 3

1–3

Which Australian time zone is closest to the UTC time?

What is the UTC offset for Australian Central Daylight Time (ACDT)?

Do these Australian states or territories observe Daylight Saving Time (ADST)? a

Victoria

Western Australia

New South Wales

Northern Territory

Queensland

South Australia

Fluency 4

4, 6ab, 7ab, 8

6ab, 7ab, 8

5, 6cd, 7cd, 8

Which location will be the ﬁrst to celebrate New Year’s Eve? • Canberra (UTC + 10) • Darwin (UTC + 9) • Perth (UTC + 8)

17.03 Australian time zones mathspace.co

771

AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

2h behind

0.5 h behind

Compared to AEST

NT QLD WA SA NSW VIC TAS

Determine the time in Perth when it is 6:42 p.m. in Hobart.

Complete the table to show the equivalent time in each state: Western Australia (WA) 6:15 ⬚ ⬚ 5:25

772

South Australia (SA) ⬚ ⬚ 10:15 ⬚

Queensland (QLD) ⬚ 11:45 ⬚ ⬚

Complete the table to show the equivalent time in each time zone: AWST ⬚ 5:11 ⬚ ⬚

ACST 5:25 ⬚ ⬚ 7:17

AEST ⬚ ⬚ 12:25 ⬚

Convert these Daylight Savings Time in Sydney to Eastern Standard Time: a

16:14 AEDT

8:00 a.m. AEDT

2:49 a.m. AEDT

11:50 p.m. AEDT

Convert these Eastern Standard Time in Sydney to Daylight Savings Time: a

14:35 AEST

4:00 a.m. AEST

9:45 a.m. AEST

5:13 p.m. AEST

Mathspace WA Curriculum – Year 7 mathspace.co

Ex 1

When the time is 19:20 in Perth determine the time in: a

Sydney during No ADST.

Sydney during ADST.

AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

ACDT (UTC + 10.5)

AEDT (UTC + 11)

2 h behind

0.5 h behind

Compared to AEST

0.5 h ahead

1 h ahead

NT QLD

QLD

WA SA

SA NSW

NSW

VIC

TAS NO DST

Reasoning

9–11

TAS DST

9–11

During daylight saving, Jasper caught a flight from Melbourne to Perth at 9 p.m. Melbourne time on a Thursday. After a 5.5 hour flight he arrived in Perth to find that it was still Thursday. Explain why this could not happen with the same length of flight if he had flown from Brisbane.

Explain why a flight from Sydney at 6:00 a.m. AEDT will arrive in Perth at 9:30 a.m. AWST after a 5-hour flight. Consider the time difference due to Daylight Saving Time.

Why does a train departing from Melbourne at 10:00 p.m. AEDT arrive in Sydney at 1:00 a.m. AEDT the next day after a 3-hour journey?

Problem-solving 12, 13ab, 14a 12

12, 13bd, 14c

13ace, 14b

Tina looks at her flight details. Her plane leaves at 6:00 a.m. on Wednesday and arrives at her destination at 8:00 p.m. on Tuesday (local time). The flight will take 12 hours. Calculate the time difference between her origin and destination.

17.03 Australian time zones mathspace.co

773

The map shows the time zones across the Australian States: AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

ACDT (UTC + 10.5)

AEDT (UTC + 11)

2 h behind

0.5 h behind

Compared to AEST

0.5 h ahead

1 h ahead

NT QLD

QLD

WA SA

SA NSW

NSW

VIC

TAS NO DST

DST

A ﬂight leaves Hobart Airport, TAS at 10:55 a.m. during ADST. It takes 4 hours and 30 minutes to reach Perth airport, WA. Determine the time the ﬂight arrives in WA local time.

A ﬂight leaves Alice Springs Airport, NT at 5:15 a.m. during ADST. It takes 3 hours and 10 minutes to reach Hobart airport, TAS. Determine the time the ﬂight arrives in TAS local time.

If an airplane leaves Broome Airport in Western Australia at 7:30 p.m. and arrives at Albury (operating on daylight savings time) Airport in Victoria at 3:52 a.m., determine its ﬂight time.

If an airplane leaves Albury Airport in Victoria at 7:14 p.m. and arrives at Adelaide Airport in South Australia (operating on daylight savings time) at 9:10 p.m., determine its ﬂight time.

If an airplane leaves Adelaide Airport in South Australia at 9:53 a.m. and arrives at Broome Airport in Western Australia at 3:41 p.m., determine its ﬂight time.

Luke and Maria board a train at 7:58 p.m. on Saturday from Sydney, New South Wales towards Adelaide, South Australia. a

If the trip to Adelaide took 20 hours and 22 mins, determine the time they arrived in Adelaide local time.

Luke boards a train from Adelaide, South Australia at 5:45 p.m. on Sunday to Darwin, Northern Territory. If his trip took 28 hours and 36 mins, determine the time he arrived in Darwin local time.

774

TAS

Maria boards a train from Adelaide, South Australia at 4:55 p.m. Sunday towards Perth, Western Australia. If the trip took her 19 hours and 20 mins, determine the time she arrived in Perth local time.

Mathspace WA Curriculum – Year 7 mathspace.co

AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

2h behind

0.5 h behind

Compared to AEST

NT QLD WA SA NSW VIC TAS

17 Chapter review 1

Describe the difference between 12 and 24-hour time.

Convert: a

7 days to hours

15 minutes to seconds

48 hours to minutes

12 hours to seconds

300 seconds to minutes

360 minutes to hours

900 seconds to minutes

600 minutes to hours

A train ride from Sydney to Canberra takes 180 minutes. How many hours and minutes is this?

Emily’s cooking class starts at 1:30 p.m. and ends at 4:00 p.m. How long is the cooking class in hours?

John starts reading a book at 8:30 a.m., which takes 2 hours and 20 minutes to ﬁnish. What time will John ﬁnish reading the book?

A software installation on a server starts at 9:15 p.m. The installation will last for 105 minutes. Find the time that the installation will be completed.

Express the times in 24-hour time:

Five to seven in the evening

Twenty-ﬁve past two in the afternoon

Half past ten at night

Quarter to eleven at night

Quarter to three in the afternoon

Five past six in the morning

Find the time and day that is represented by: a

21 hours and 30 minutes after 19:45 on Monday

16 hours and 45 minutes before 12:30 on Friday

18 hours and 40 minutes after 20:15 on Sunday

24 hours and 20 minutes before 13:55 on Tuesday

Sarah and her family set out for a day trip to the beach. They departed at 3:15 p.m. State this time in 24-hour time.

Mike starts grilling steaks for a dinner party. The steaks take 45 minutes to cook. Mike puts them on the grill at 4:35 p.m. What time did Mike take the steaks off the grill? Express your answer in 24-hour time.

A ﬂight between two cities takes 7 hours and 15 minutes. If you leave one city at 21:10, at what time will you arrive at the other city? Express your answer in 24-hour time.

If an average person reads for 20 minutes per day, ﬁnd how many hours they spend reading in: a

One year (assume non-leap year).

40 years (ignoring leap-years).

Chapter 17 review mathspace.co

775

A day on Jupiter is 9.9 Earth hours and a year on Jupiter is 4332.59 Earth days. How long is a year on Jupiter in Jupiter days? Give your answer to the nearest Jupiter day.

Today is Alice’s birthday and she just turned 12 years old. Assume there are 365 days in a year. Find her age in: a

Days

Hours

Minutes

Seconds

Put the times in order from earliest in the day to latest in the day: 02:15, 18:45, 12:45 p.m., 12:45 a.m., 10:30 p.m., 16:30

Do these Australian states or territories observe Daylight Saving Time (DST): a

Western Australia

New South Wales

Tasmania

South Australia

Convert the Daylight Savings Time in Sydney to Eastern Standard Time: a

2:49 a.m. AEDT

11:50 p.m. AEDT

19:43 AEDT

22:56 AEDT

When the time is 18:15 in Perth determine the time in: a

Sydney during No DST

Sydney during DST

AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

ACDT (UTC + 10.5)

AEDT (UTC + 11)

2 h behind

0.5 h behind

Compared to AEST

0.5 h ahead

1 h ahead

NT QLD

QLD

WA SA

SA NSW

NSW

VIC

TAS NO DST

776

Mathspace WA Curriculum – Year 7 mathspace.co

TAS DST

“Time is only a vague concept for us; we never know what we mean by it. We only know it indirectly through movement, and movement is not always a trustworthy measure.” Henri Poincaré

Answers

2 B y adding the products which are inside each portion.

1.01 Addition and subtraction

12 × 8 = 8 × 10 + 8 × 2 = 80 + 16

Understanding

= 96

1 a Addition

b Addition

c Subtraction

d Addition

e Subtraction f Subtraction

g Addition

h Subtraction i Addition

j Subtraction

k Addition

Fluency 3 a

l Subtraction

2 a 8858

b 2838

c 86

d 528

e 1714

f 750

g 482

h 1485

b 990

c 791

d 9923

e 6679

f 64 695

g 59 240

h 1500

i 957

j 1726

k 13 643

l 11 842

4 a 99 699

b 64 770

c 41 415

d 3607

e 2373

f 91 729

g 59 662

h 8468

4000

150

b 4150

Fluency 3 a 99

4 a 6

3600

48000

4800

1800

180

6 342 + 35 = 300 + 40 + 2 + 30 + 5 = 300 + 77 b True

e True

f True

c True

d False

8 A = 9, B = 0 and C = 7 Problem-solving 9 80 pieces of fruit

b 54 780

600

5 18 + 12 = 10 + 8 + 10 + 2 = 20 + 10

7 a False

5 a

i 832 Reasoning

b 3648

600 8

6 a 900

72000

5600 320

6300

490

b 84 738

7 a 6

b 7

c 36

d 8

10 341 species

e 25

f 12

g 21

h 30

11 130 seconds

i 30

j 77

k 36

l 60

8 a 180

12 30 bricks

1.02 Multiplication Understanding 1

3 5 15 tiles 3 × 5 = 15 15 tiles 5 × 3 = 15

778

Mathspace WA Curriculum – Year 7 mathspace.co

b 270

c 261

d 351

e 3888

f 2367

g 2268

h 4380

i 16 665

j 11 106

k 2596

l 10 707

9 a 280 000

b 30 000

c 1 200 000

d 21 000

e 3978

f 15 594

g 12 630

h 10 374

i 23 446

j 46 397

k 246 666

l 475 626

10 a 90 e 120

b 48

c 630

d 48

f 1200

g 240

h 1125

Reasoning

11 Sample answer: 5 × 6 is the same as 5 × 2 × 3, which is the same as 10 × 3. 12 a 8 × 70 + 8 × 5 c 80 × 64 + 2 × 64

b 11 × 90 + 11 × 2 d 70 × 475 + 3 × 475

Problem-solving 13 a 2 and 3

b 2 and 12

c 9 and 4

d 36 and 36

14 a 45 hours

b Tristan

16 $29 190 17 $542 500 18 904 320

9 3×9+1

Problem-solving

19 $56 128 960

10 To solve this problem, we need to find a number that leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 7. One way to do this is to start with the number 30 and add 1 until we find a number that satisfies both conditions.

20 23 760

1.03 Division Understanding

2 a False

8 Dividing by zero is an invalid operation in mathematics that leads to undeﬁned behaviour, causing problems in mathematical models. Therefore, it is crucial to avoid undeﬁned values to maintain the consistency and validity of our mathematical systems. In this expression, 3 is the divisor, 9 is the quotient, and 1 is the remainder. When you multiply 3 by 9, you get 27. Then, adding the remainder gives you 28, which is the original dividend, confirming the correctness of the rewrite.

15 530 m

1 a 25 ÷ 5

7 x ÷ y = y ÷ x when x = y or x = −y and both x and y are not equal to 0.

b 24 ÷ 8

c 49 ÷ 7

When we divide 30 by 3, we get a remainder of 0. Therefore, the next number that leaves a remainder of 2 when divided by 3 is 30 + 2 = 32.

b False

c False

d True

b 18

c 30

d 90

e 91

f 276

g 260

h 257

i 749

j 2393

k 907

l 1023

4 a 40

b 62

c 74

d 120

e 45

f 44

g 314

h 299

11 16 biscuits

i 666

j 606

k 605

l 600

12 50 glasses

Fluency 3 a 9

When we divide 32 by 7, we get a remainder of 4. Therefore, 32 is the smallest number greater than 30 that leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 7.

5 a 177 r. 3

b 781 r. 2

c 821 r. 3

13 17 boxes

d 925 r. 5

e 680 r. 5

f 1019 r. 1

14 44 canned items

g 914 r. 6

h 1862 r. 2

i 3078 r. 2

j 4793 r. 3

k 3734 r. 4

l 10 532 r. 1

b 7 r. 28

c 13 r. 13

d 22 r. 5

e 107 r. 23

f 106 r. 4

g 288 r. 10

h 540 r. 1

i 499 r. 20

j 490 r. 16

k 614 r. 1

l 1286 r. 17

6 a 24 r. 3

15 78 rows 16 140 plants 17 a 476 ÷ 17 = 28

b 434 ÷ 31 = 14

c 560 ÷ 14 = 40

d 651 ÷ 21 = 31

e 816 ÷ 34 = 24

f 837 ÷ 31 = 27

g 924 ÷ 21 = 44

h 999 ÷ 37 = 27

Answers mathspace.co

779

1.04 Order of operations with whole numbers

10 a i No

Understanding

ii Yes

iii No

iv Yes

b The addition is now done before the division.

1 The order of operations when performing arithmetic with whole numbers is as follows: 1. First, perform any calculations inside parentheses or brackets, working from the innermost set of parentheses to the outermost. 2. Next, perform any multiplication or division operations from left to right. 3. Finally, perform any addition or subtraction operations from left to right. 2 a 3×5

b They cause the addition to be evaluated before the multiplication.

b 4×2

c 9÷3

d 7−2

11 a i Yes

ii No

iii No

b The subtraction is now done before the multiplication. 12 a i No

ii Yes

iii No

b The subtraction is now done before the division. Problem-solving 13 a ×, −

b ×, ×

c ÷, ×

d +, ×

e ×, +

f ÷, −

14 50 apples

Fluency

15 $22

3 a 14

b 16

c 14

d 16

e 19

f 40

g 78

h 16

16 2 lollies

i 34

j 13

k 27

l 5

n 10

17 22 marbles

4 a 160

b 13

c 1

d 48

e 20

f 60

g 58

h 4

1.05 Check reasonableness of calculations

i 12

j 11

k 42

l 1

n 4

5 a 13 b i 51

ii 13

iii 63

ii 63

iii 88

6 a 88 b i 82 7 a 19 b i 19

Understanding 1 a 20

b 80

c 60

d 350

e 130

f 100

g 80

h 910

2 a 800

b 300

c 900

d 100

e 200

f 700

g 500

h 1000

3 a 1000

b 7000

c 9000

d 5000

f 45 000

g 19 000

h 67 000

e 10 000 ii 15

iii 89

iv 5

8 a 12 − (6 + 6) ÷ 6 = 10

4 • “The product is less than 4000.” is correct. • “The product is more than 2700.” is correct.

b (5 + 10) × 4 – 5 × 4 = 40

• “The product is more than 4000.” is incorrect. The product is less than 4000.

c 4 + 5 × (13 – 5) – 4 = 40 d 5 × 15 − (4 ÷ 2 + 8) = 65

• “The product is less than 2700.” is incorrect. The product is more than 2700.

e 18 × (4 − 1) + 4 − 18 = 40 f (23 − 10) × 3 + 8 × 9 – 2 = 109 g 15 ÷ 3 + 7 × (4 + 2) × 6 = 257

Fluency

h 30 − 4 × 5 ÷ (5 – 3) = 20

5 a Yes, approximately 60 − 30 = 30 b No, approximately 70 − 30 = 40

Reasoning 9 a i No

780

ii Yes

iii No

Mathspace WA Curriculum – Year 7 mathspace.co

iv Yes

c Yes, approximately 2880 − 2830 = 80 − 30 if we subtract 2800 from both numbers.

d No, approximately 6870 − 5830 and if we subtract 5800 from both values, this equals to 1070 − 30 not 70 − 30. 6 a No, 4 × 80 = 320, since 89 > 80 it will be more than 320. b Yes, 240 = 3 × 80 and since 85 is great than 80, so this answer will be larger than 240.

b Yes, because then they will have $22.40 which will cover the cost of two pizzas. 25 Example answer: To find the approximate area of the sticker, you can round the length by the width first to the nearest tens and then multiply them. The dimensions are approximately 60 mm and 110 mm. The approximate area is 60 mm × 110 mm = 6600 mm2.

7 a 10

b 100

c 1000

d 1000

8 a Yes

b Yes

c No

d Yes

9 a Yes

b Yes

c No

d No

10 a Yes

b No

c No

d Yes

11 a Yes

b No

c Yes

d No

b 900

c 400

d 500

1.06 Number calculations with technology

d Larger

Understanding

12 No 13 No 14 a 4200 Reasoning 15 a Larger

b Smaller c Larger

26 Example answer: To find the approximate number of students in the school, you can round the number of students to the nearest tens and then add them. The approximate number of students are 120 for Year 7, 190 for Year 8, 210 for Year 9 and 50 for Year 10. The approximate number of students in the school is 120 + 190 + 210 + 50 = 570.

16 a Yes, the estimate will be larger.

1 B

b Yes, the estimate will be smaller.

2 B

c No

3 A

17 a Smaller b Larger

c Smaller d Larger

18 a No

b No

4 A

c Yes, the estimate will be larger.

Fluency

d Yes, the estimate will be smaller.

5 a 514

19 a Smaller b Larger

c Larger

d Larger

d 80

f 509

g 79

h 26 096

c 10 441

d 80

e 490 260

b Yes, the estimate will be smaller. c No

22 a No

c 1172

6 a 720 000 b 9

20 a Yes, the estimate will be larger.

21 a Smaller b Smaller c Larger

e 135 194

b 61 809

d Larger

b No

c Yes, the estimate will be larger. d Yes, the estimate will be smaller.

7 a False

b False

c True

d False

e False

f True

g False

h True

i False

j False

Reasoning 8 No, he swapped the digits in a number.

Problem-solving

9 Yes

23 7 boxes that hold 30 padlocks each.

10 No, he forgot a zero.

24 a No, because he is rounding too much. Two pizzas cost $19.50, so Michael cannot afford two pizzas.

11 No, he added an extra zero. 12 No, he swapped some of the numbers. 13 Yes

Answers mathspace.co

781

14 No, he didn’t use the correct order of operations.

Reasoning 13 a Example answer: 15 apples were initially in a box, and then 2 groups of 4 apples were removed.

Problem-solving 15 28

b Example answer:

16 a 6

b $384

10 cookies were divided into 2 groups, and 3 more cookies were added.

17 9 lollies 18 Alﬁe gets 20 marbles, Barry gets 40 marbles and Corey gets 60 marbles 19 3

1.07 Write mathematical statements Understanding 1 a More than, add, plus, sum, increase, total b Less than, subtract, minus, difference, decrease, take away, deduct c Multiply, times, product, groups of

c Example answer: There were 10 cupcakes, but 4 of them fell onto the floor and had to be discarded. The amount left over were then shared between two people. d Example answer: 8 pencils were initially in the box, and 3 groups of 2 pencils were added to the box. Problem-solving 14 Maximum Kilometres = (425 − (5 × 30)) ÷ 0.25 15 (21 × 76) − (20 × 75)

d Divide, split, quotient, share, ratio 2 a x

b vi

c v

d ix

16 (12 − 4) ÷ 2 + 4

e ii

f i

g viii

h vii

17 (20 ÷ 4) × (20 ÷ 4)

i iv

j iii

Chapter 1 review Fluency

1 a 7 × 3 + (9 − 12) + 2 = 20

3 a 2+7

b 7−6

c 3 × 10

d 12 ÷ 4

b (5 + 10) × 4 − 5 × 4 = 40

e 4×7

f 13 ≠ 11

g 4 < 10

h 8>5

c (9 + 6) ÷ 3 × 4 − 8 = 12

4 a 15 = 7 + 8 c 80 = 8 × 10

b 6 = 16 − 10 d 5 = 50 ÷ 10

e 19 > 7 + 5

2 a 9 b i 20

5 a

b 12 × (10 − 6)

d 11 × 2 − 9 ≠ 10

f 20 ≥ 2(3 + 6)

3 a +, ×

ii 9

iii 24

b ×, +

c ×, ÷

d +, ×

b 9000

c 7000

d 2000

f 44 000

g 18 000

h 70 000

4 $75 5 a 1000 e 10 000

6 a Yes, approximately 80 + 30 = 110

6 C

b No, approximately 80 + 20 = 100. The actual addition is 76 + 15 = 91. The statement is not reasonable because the estimated result 100 is significantly different from the actual result 91.

7 D 8 B 9 A

c Yes, approximately 9000 + 3000 = 12 000. The actual addition is 9134 + 2678 = 11 812 which is close to the actual result.

10 A 11 D 12 a 2 + 80 ÷ 10

782

d (14 ÷ 2) × 4 + 9 − 3 × 2 = 31

b 10

Mathspace WA Curriculum – Year 7 mathspace.co

d No, we can round both numbers to the nearest thousand and perform the subtraction: 7500 − 1000 = 6500. The actual subtraction is 7652 − 1389 = 6263. The statement is unreasonable because the estimated result (6500) is not close enough to the actual result (6263) to be a reasonable approximation. A better estimation would be to round the numbers to the nearest hundred: 7700 − 1400 = 6300, which is much closer to the actual result.

2.01 Number operation properties Understanding 1 a Associative law

b Reordering

c Commutative law 2 Answer may look like:

7 a Yes

b Yes

c No

d No

The distributive law allows us to break apart larger numbers to help us calculate larger multiplications or divisions. For example: 1326 ÷ 13 = (1300 ÷ 13) + (26 ÷ 13).

8 a 10

b 100

c 1000

d 1000

3 a Commutative law, reordering

9 a Yes

b No

c Yes

d Yes

10 a 4500

b 1000

c 400

d 500

b Commutative law, reordering c Reordering d Associative law e Distributive law

11 A 12 a False

b True

c True

d False

e True

f True

g False

h False

13 29

f Distributive law g Reordering h Reordering 4 a True

14 a 3

b False

c False

d True

b $768 Fluency

15 12 lollies 16 a 2 + 7

b 7−6

c 3 × 10

d 12 ÷ 4

e 4×7

f 13 ≠ 11

g 4 < 10

h 8>5

17 a The sum of 12 and 8 is equal to 20. b The difference between 8 and 1 is equal to 7.

5 B 6 a 75

b 58

7 a 2×5 b 2 × 16 × 5 = 2 × 5 × 16 or 2 × 16 × 5 = 5 × 2 × 16

c The product of 2 and 7 is equal to 14.

2 × 16 × 5 = 160

d The quotient of 36 and 4 is equal to 9.

8 260

e The quotient of 10 and 5 is not equal to the quotient of 5 and 10.

9 78

18 a

b 12 × (10 − 6)

10 79 11 2

d 11 × 2 − 9 ≠ 10

f 20 ≥ 2(3 + 6)

13 11

b 10

14 a 54 + 26

19 a 2 + 80 ÷ 10

12 a 26

b 12

b 118

20 (12 − 4) ÷ 2 + 4

15 340

21 (20 ÷ 4) × (20 ÷ 4)

16 a 3

b 20

c 805

22 Maximum Kilometres = (325 − (8 × 25)) ÷ 0.25

17 a 30

b 1

c 754

18 a 8982

b 7014

c 1364

d 2472

19 132 ÷ 4 ÷ 11 = 3

Answers mathspace.co

783

20 a 12

b 12

c 32

4 Answer may look like: A square number is a number that is the result of multiplying an integer by itself.

21 a 21

b 7

c 17

22 a 202

b 99

c 87

d 45

Reasoning

A cube number is a number that is the result of multiplying an integer by itself two more times. 5 Answer may look like:

23 An example answer: Consider the example 15 ÷ 5. This division problem is asking “how many groups of 5 are there in 15?” The answer to this question is 3. If we applied the commutative law, we would get the expression 5 ÷ 15 which is the same as asking “how many groups of 15 are there in 5?” This would not be a whole number, which shows 15 ÷ 5 ≠ 5 ÷ 15, so the commutative law does not work for division. 24 In step 2, Julietta applied the associative law, but the associative law does not work for subtraction.

We can determine if a number is a square number by looking at its factors. If it has two factors that are the same, then it is a square number. Fluency 6 a 49

b 144

c 529

d 40 000

7 a 27

b 125

c 2197

d 64 000

8 a 58

b 10 100

c 17

d 9900

e 4

f 1 000 000

g 16

h 64

25 Regroup the problem so that it is 5 × 200 × 34 which is 1000 × 34 = 34 000

9 a 14

b 52

c 25

d 40

e 26

f 200

g 5

h 5

Problem-solving

10 a 9

b 72

c 19

d 98

f 27 952

g 27

h 8

11 a 5

b 80

c 144

d 3

e 25

f 625

g 1

h 217

26 a 160

e 512

b 16 d 8 × 20 + 8 × 2

c 176 27 a 320

b 32 d 16 × 18 + 16 × 2

c 288

28 Cost of the shirts: 4 × $15 = 60

13 324 chairs

Cost of the pants: 3 × $25 = 75 Calculating the discount:

Reasoning

0.10 × (60 + 75) = 0.10 × 60 + 0.10 × 75

12 10 648 cm2

Distributive law

= 6.00 + 7.50

Evaluate the multiplication

= 13.50

Evaluate the addition

2.02 Square and cube numbers

14 Answer may look like: It is not possible to do this with whole number side lengths. A non-square whole number is the product of two different numbers. If a square is formed, the side lengths would be decimals. 15 a 1000 – 343 = 657 b 27

Understanding

c No, since 103 – 73 ≠ (10 – 7)3.

1 a B

a C

2 a 9

b 2

b 216

3 a Neither

b Perfect cube

c Yes, since 23 × 33 = (2 × 3)3.

c Perfect square

d Both

d 512

e Neither

f Perfect square

g Perfect cube

h Both

784

Mathspace WA Curriculum – Year 7 mathspace.co

16 a 8 × 27 = 216

17 a 36 ÷ 4 = 9 b 9

22 a

c Yes, since 62 ÷ 22 = (6 ÷ 2)2.

Number

d 16 18 a 16 + 25 = 41 b 81 c No, since 42 + 52 ≠ (4 + 5)2. 19 Answer may look like: Odd numbers will always have 1, 3, 5, 7 or 9 in the units place value. When these numbers are squared, the units place value will be 1, 9, 5, 9, 1 respectively, which are all odd digits. Problem-solving

Cube of number

Final digit of cube number

125

216

343

512

729

1000

Number

Cube of number

Final digit of cube number

1331

1728

2197

20 81 cm

21 a i 1

ii 3

iii 5

iv 7

b We obtain the sequence of odd numbers: 1, 3, 5, 7, 9, … c Example answer: To grow the square at each stage we need to add on a new L-shaped layer. 9 7 5 3 1 1

4 +3

9 +5

16 +7

25 +9

This layer can be viewed as adding two lots of the previous side length plus one corner square. Previous side length

Previous side length

This pattern increases by two at each stage starting with one. This makes a sequence of odd numbers.

2744

3375

4096

4913

5832

6859

8000

c The pattern is a repeating sequence 1, 8, 7, 4, 5, 6, 3, 2, 9, 0. d The final digits of the cube numbers from 21 to 30 are 1, 8, 7, 4, 5, 6, 3, 2, 9, 0 respectively. e Yes, because when you cube a number, the digit in the units place is the only one that determines the final digit of the cube number. Digits in any position other than the units place (e.g., tens place) do not affect the final digit. So, when we cube the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, the final digits of the cube numbers will always be 1, 8, 7, 4, 5, 6, 3, 2, 9, 0 respectively. 23 4 24 25 25 106 26 52

Answers mathspace.co

785

2.03 Square and cube roots

Reasoning .

18 Example:

Understanding 1 a 64 = 8 × 8, therefore the square root of 64 is 8. b 64 = 4 × 4 × 4, therefore the cube root of 64 is 4. 2 Answer may look like: They are inverse operations. That is, if you take the square root of a number and then square the result, you get the original number. 3 I f the number is a perfect cube, the cube root is a whole number, and if it is not a perfect cube, the cube root is not a whole number. 4 a No

b Yes

c Yes

d No

e Yes

f No

g Yes

h Yes

5 a True

b False

c False

d True

Fluency

Justification: 43 = 64 and 53 = 125, so . 64 < 80 < 125, meaning 19 Answer may look like: One way to estimate the value of a square root is to find the perfect square that is closest to the number and then estimate the square root based on its distance from that perfect square. For example, the square root of 53 is between the perfect squares of 49 and 64, so we can estimate it to be between 7 and 8. 20 Answer may look like: , but Ansha correctly identified that her mistake was saying the square root is is closer approximately 6. This is because to 7, so it should be approximated as 7. We is closer to 7 because 49 − 43 = 6 know but 43 − 36 = 7. 21 a Answers may look like:

6 B

Donny took the square root of 64 instead of taking the cube root.

7 C 8 a 2

b 4

c 7

d 12

9 a 20

b 8

c 60

d 2

e 11

f 10

g 5

h 1

i 242

j 139

k 21

l 5

10 9 m

He divided 27 by 3 instead of taking its cube root. b

22 a D b Answer may look like:

12 a 16 < 22 < 25

13 a 3 and 4

b 2 and 3

and is closer to 3 because 11 − 9 = 2 while 16 − 11 = 5. Because the is difference between 11 and 9 is smaller, = 3. closer to

c 5 and 6

d 9 and 10

e 7 and 8

f 11 and 12

23 a C

g 15 and 16

h 20 and 21

11 125 m by 125 m

14 a 4

b 7

c 8

d 11

15 a 5

b 6

c 3

d 14

e 2

f 8

g 1

h 16

i 147

j 42

k 5

l 4

b Answer may look like: is closer to 6 because and 200 − 125 = 75 while 216 − 200 = 16. Because the difference is between 200 and 216 is smaller, = 6. closer to

Problem-solving

16 4 17 7 m

24 a 10 25 168 cm 26 1600 cm

786

Mathspace WA Curriculum – Year 7 mathspace.co

b 10

c Yes

d 12

27 a

Shape number (n)

g 122 × 93

h 112 × 73

Total number of tiles (T )

i 123 × 152

j 22 × 42

b The total number of tiles (T ) is equal to the square of the shape number (n) that is added to 1. c

Shape number (n)

Number of light purple border tiles (B)

2 8

9 a 9×9

b 9×9×9

c 9×9×9×9

d 9×9×9×9×9

e 3×3×3×3×3

f 5×5×5×3

g 2×2×2×2×2×2×2×2

d The number of light purple border tiles (B) is equal to 4 times the shape number (n).

l 22 × 33 × 52

k 24 × 12

h 7×7×7×7×7×7 10 a 5 × 5 b 7×7×7×7×7×6×6×6×6

e Shape number 6.

f 100 tiles

c 4×4×4×4×4×4×9×9

g 64 tiles

h 32 tiles

d 77 × 77 × 77 × 77 × 77 × 13 × 13

28 a 3 ft

b 2 ft

11 a 5 × 5 × 5

2.04 Powers of whole numbers

Understanding

b 125

128

256

512

13 a 1, 3, 9, 27, 81, 243, 729

1 9

b 1, 5, 25, 125, 625

2 7

Reasoning

3 a The value of the expression increases. b The value of the expression decreases.

14 As the base is 4 and the power is 5, you multiply 4 by itself 5 times. So 45 = 4 × 4 × 4 × 4 × 4

4 C

15 a Fluency 5 a 22

c 52

d 23

Words

Expanded notation

Index notation

Five to the power of six

5×5×5×5× 5×5

6×6×6×6

Eight cubed

8×8×8

Eleven to the power of three

11 × 11 × 11

113

Twenty three to the power of five

23 × 23 × 23 × 23 × 23

235

Seven squared

7×7

Fifteen to the power of four

15 × 15 × 15 × 15

154

b 33

Six to the power of four

7 a 92

b 93

c 94

8 a 53

b 16

c 34

d 24

e 4

f 7

d 95

b 1 when raised to any power will always be 1. 16 a The answer is halved. c i

iii

b 1

125

625

100

101

102

103

104

100

1000

10 000

d Every non-zero number raised to the power of zero equals one. Problem-solving 17 a 0

b 5

c 3

d 1

e 4

f 4

g 9

h 6

18 a 64

b 1

c 116

d 243

e 128

f 9

g 68

h 130

Answers mathspace.co

787

i 140 19 a >

j 98

k 129

l −37

Reasoning

b =

c >

d =

11 a > b 12 9 × 104 because it only has 4 place values while 2 × 105 has 5 place values. If we write both numbers as whole numbers, we get 90 000 and 200 000. This shows 9 × 104 is smaller.

20 81

2.05 Powers of ten Understanding

13 Answer may look like:

1 12 zeros

Multiplying by 10 increases the value of a number by a factor of 10, while raising to the power of 10 multiplies the number by itself 10 times.

2 D 3 105

For example, 2 × 10 = 20 but 210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

Fluency 4

101

14 Answer may look like:

100

We can use powers of ten to compare numbers by writing them in expanded form, then comparing the powers on the bases of ten. If the powers on the bases of ten are the same, then we can compare the digits that are multiplied to the powers of ten.

103

1000

104

10 000

105

100 000

1 000 000

107

10 000 000

108

100 000 000

Problem-solving

5 6

15 384 000 kilometres

6 a 200 000

b 300

c 7000

d 10 000

e 80

f 6 000 000

17 a 50 300 000 km

g 3

h 50 000 000

b 41 400 000 km

7 a 102 e 6 × 103

16 171 300 000 kilograms

b 103

c 4 × 102

d 9 × 102

f 7 × 103

g 8 × 104

h 3 × 104

8 500 000 000 kilometres

2.06 Factors and multiples Understanding

9 a 2 × 103 + 2 × 102

1 Answer may look like:

b 8 × 103 + 3 × 102 c 2 × 103 + 4 × 102 + 8 × 101 d 7 × 103 + 2 × 101 e 6 × 103 + 2 × 102 + 6 × 101 f 9 × 104 + 2 × 103 g 3 × 104 + 3 × 103 + 5 × 101 h 6 × 104 + 2 × 103 + 4 × 102 + 8 × 101 10 a 37 000

b 1040

c 410

d 7640

e 60 305

f 32 095

g 324 020

h 22 776

788

c The distance between Venus and Earth

Mathspace WA Curriculum – Year 7 mathspace.co

A factor of a number is a whole number that divides that number exactly without any remainder. 2 Answer may look like: A factor of a number divides the number exactly, whereas a multiple of a number is obtained by multiplying the number by another whole number. Factors will be smaller or equal to the number itself, whereas multiples will be equal or larger than the number itself. 3 1

4 2

15 a

Week

5 A number is a multiple of another number if that other number divides into it evenly.

120

150

180

Fluency 6

Factor pairs of 50 (1, 50) (2, 25)

Total earnings (in dollars)

b $150

(5, 10) 7 a Yes

b No

c No

d Yes

Reasoning

e Yes

f No

g Yes

h No

16 For a number to have exactly three factors, it must be a square of a prime number. This is because a prime number has only two factors (1 and itself), so the only way for a number to have exactly three factors is for it to be the product of two identical prime numbers.

8 a 1, 2, 7, 14 b 1, 19 c 1, 3, 9 d 1, 3, 11, 33 e 1, 2, 4, 7, 14, 28 f 1, 2, 4, 8

17 Answer may be like:

g 1, 2, 3, 6, 7, 14, 21, 42

Yes, the number itself is both a factor and a multiple of itself.

h 1, 2, 4, 8, 16 i 1, 3, 5, 9, 15, 45

18 This is possible because 35 is a factor of 280.

j 1, 7, 49

19 a 6

k 1, 2, 3, 6, 9, 18, 27, 54 l 1, 2, 3, 6, 8, 9, 12, 24, 36, 72 9 a 1, 5, 23, 115

20 a 1, 2, 4, 8, 16

b 1, 2, 7, 14, 49, 98 c 1, 3, 7, 11, 21, 33, 77, 231 d 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, 312 10 a 4, 8, 12, 16, 20

b 3, 6, 9, 12, 15

b The number of ﬂowers that can be in each row are the factors of 16. Problem-solving 21 108 eggs

c 7, 14, 21, 28, 35

d 9, 18, 27, 36, 45

e 10, 20, 30, 40, 50

f 11, 22, 33, 44, 55

g 12, 24, 36, 48, 60

h 15, 30, 45, 60, 75

23 72

c No

24 Mindt

11 a Yes

b No

b 4

c There are marbles left over because 6 is not a factor of 40.

d Yes

22 3

12 32

25 a 36 ﬁsh

13 80

26 a 15 rooms

14 6

27 16 m by 24 m

b 20 ﬁsh

c Wally

b 18 weeks

2.07 Prime factorisations Understanding 1 a Factor of a number

b Prime number

c Prime factorisation of a number

Answers mathspace.co

789

2 a Prime: 3, 5

ii 2 × 32 × 7

e i

Composite: 4, 6, 10, 12

126

b Prime: 7, 13, 11 6

Composite: 14, 15 c Prime: 19, 17

Composite: 1, 21, 27 3 Factors and prime factors are the same in that they both divide the number evenly. They are different because factors can be prime numbers, composite numbers, or neither, such as 1. Prime factors are always prime numbers.

144

2 18

ii 23 × 52

g i 200

5 45 2

ii 2 × 3

b 2 ×3 ×5×7

6 a 1260 7 a i

ii 24 × 32

f i

Fluency 4

ii 23 × 32 × 5

h i 2

b i

360

ii 22 × 7

c i

ii 2 × 3 × 7 7

d i

2 42

8 a 2×2×3×3

ii 34

d 3×2×2×2×2 f 2×2×2×2×2×3

g 2 × 2 × 3 × 11

h 2×3×5×5

9 a 3 ×5×2

b 52 × 23

c 22 × 32 × 5

d 52 × 32

f 24 × 52

e 2 × 13 g 7×5×2

Reasoning 3

790

Mathspace WA Curriculum – Year 7 mathspace.co

b 2×3×7

e 2×2×2×3×3

c 2×3×3×3

10 Answer may look like:

h 23 × 32 × 7

No, two different numbers cannot have the same prime factorisation. This is because the product of the prime factors will result in the same number. We cannot multiply the same numbers together and get two different answers. 11 Rae and Josie are both correct. 2 and 5 are both prime factors of 30, so they will need to divide by both numbers at some point in their work. It does not matter which one is divided ﬁrst as either way will lead to the same ﬁnal result. 12 The prime factors of 35 are 5 and 7. If both 5 and 7 divide 420 evenly, then 35 is a factor of 420. Problem-solving 13 a No, 175 = 5 × 5 × 7

2.08 HCF and LCM Understanding 1 a Highest common factor (HCF) is the largest positive integer that divides two or more integers without a remainder. b Least common multiple is the smallest positive integer that is a multiple of two or more integers. 2 a True b False. If the HCF and LCM of two numbers are equal, then the numbers are equal. c True d False. The HCF of two numbers cannot be larger than either of the two numbers because a factor of a number cannot be greater than the number itself. e False. The LCM of two numbers cannot be less than either of the two numbers because a multiple of a number must be greater than or equal to the number itself.

b Yes c No, 286 = 2 × 11 × 13 d No, 27 = 3 × 3 × 3 e No, 20 = 2 × 2 × 5 f No, 189 = 3 × 3 × 3 × 7

Fluency 3 a i 1, 2, 3, 6

14 a No, 56 = 23 × 7 b No, 495 = 3 × 3 × 5 × 11 c Yes

1, 2, 3, 4, 6, 12 ii 6 b i 1, 3, 9

d Yes e No, 504 = 23 × 32 × 7 2

f No, 300 = 2 × 3 × 5

1, 2, 3, 4, 6, 8, 12, 24 ii 3 c i 1, 2, 7, 14

15 36

1, 2, 4, 8, 16, 32

16 a The ﬁnal number is the same as the original number.

ii 2

b All six-digit numbers where the ﬁrst three digits are the same as the last three digits have 7, 11 and 13 as prime factors. c 1001 = 1000 + 1 d If we multiply 1001 by any three-digit number, it is the same as multiplying by 1000 + 1. For example, 123(1000 + 1) = 123 000 + 123 = 123 123. In other words, multiplying it by 1000 makes the ﬁrst 3 digits the same as the original number, then multiplying by 1 makes the last 3 digits the same as the original number.

d i 1, 2, 4, 7, 14, 28 1, 2, 3, 6, 7, 14, 21, 42 ii 14 4 60 5 28 6 350 7 a 8

b 6

c 3

d 12

e 6

f 36

g 11

h 10

i 16

j 24

8 a 2940

b 8820

9 a i 6, 12, 18, 24, 30, 36, 42, 48

Answers mathspace.co

791

10, 20, 30, 40, 50, 60, 70, 80

24 6 minutes

ii 30

Chapter 2 review

b i 8, 16, 24, 32, 40, 48, 56, 64 14, 28, 42, 56, 70, 84, 98, 112

1 a 27, 18

ii 56 c i 24, 48, 72, 96, 120, 144, 168, 192 42, 84, 126, 168, 210, 252, 294, 336 ii 168 d i 15, 30, 45, 60, 75, 90, 105, 120 30, 60, 90, 120, 150, 180, 210, 240 ii 30

2 a 225

b 324

c 676

d 6400

e 14 500

f 27

g 80

h 231

i 32

j 867

k 7

l 13

3 a 64

b 343

c 729

d 1728

e 973

f 728

g 125

h 24

i 333

j 46

k 24

l 15

10 108

4 625 chairs

11 3780

5 46

12 a 42 e 1872

b 52, 24

b 126

c 168

d 78

6 63

f 5850

g 72

h 378

7 a 9

b 13

c 5

d 11

13 a 75

b 34 650

e 1

f 147

g 21

h 4

14 a 17 640

b 1050

8 a 4

b 7

c 6

d 8

15 a 4 centimetres

b 25 strips

e 2

f 405

g 10

h 6

16 a Every 76 years

b Every 60 years

9 a 64 < 70 < 81

b 8<

17 a Every 126 minutes

b A

10 a 8 and 9

b 9 and 10

c 11 and 12

d 17 and 18

Reasoning

11 36 m

18 The student is correct because all numbers have 1 as a factor. If the numbers have no other factors in common, they will still have the factor of 1 in common. Any two numbers can be multiplied together to create a common multiple. 19 The product of the prime factors that the two numbers have in common will be the highest factor the numbers have in common. 20 a Sasha is not correct. Although the highest common factor of 8 and 12 is 4, the lowest common multiple is 24. b Suri is correct because both numbers have 4 as a factor. Since 4 is a factor, both numbers will be multiples of 4. c 12, 16 or 4, 48 Problem-solving 21 28 cm

792

13 192 cm 14 a 64 3

b 152 × 73 2

d 22 × 33 × 52

c 16 × 20

15 a 4 × 4 × 4 × 4 × 4 b 5×5×5×5 c 2×2×2×2×2×2×2×2 d 7×7×7×7×7×7 e 7×7×7×7×7×6×6×6×6 f 4 × 4 × 4 × 4 × 4 × 4 × 9 × 9 g 77 × 77 × 77 × 77 × 13 × 13 h 120 × 120 × 120 × 120 × 120 × 12 × 12 16 a 12 × 12 × 12 × 12

b 20 736

17 a 729

b 1024

c 207

d 432

e 704

f 512

g 6689

h 473

18 192 rabbits

22 Year 2214 23 a 18 seconds

12 4000 m

19 a 300 000 b 8 red balls

Mathspace WA Curriculum – Year 7 mathspace.co

c 7 000 000

b 800 000 d 90 000 000

20 a 2 × 103 + 2 × 102 4

c 5 × 10 21 a 530

b 8 × 103 + 4 × 102 d 6 × 10 + 9 × 10

b 8620

c 32 073

7 a

d 93 874

−10

−5

−10

−5

−10

−5

−10

−5

−10

−5

−10

−5

b c

22 2826 23 a 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

b 1, 2, 4, 8, 16, 32, 64 c 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 d 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

e f

24 60 25 a 2 × 3 × 3 × 3

b 2×2×3×5

c 2×3×3×5

d 2×3×5×5

g −15 −10

−5

−15 −10

−5

26 24 27 a 45

b 20 790

8 9

28 a 1260

b 5250

9 −1

29 30 cm

10 −8

3.01 Integers and the number line

11 −2 12 −1

Understanding 1 a Integer

13 7

b Integer

c Non-integer

d Non-integer

e Non-integer

f Integer

g Integer

h Non-integer

2 a To the right of 0

b To the left of 0 b −1, −2, −3

3 a 1, 2, 3 4 Neither

14 a 9

b 9

e −1

f 7

Example: The integers 3 and −3 are opposite integers.

b i −12

ii 12

c i 14

ii −14

d i −5

ii 5

e 10

b −3

c 2

d 9

e −2

f −9

g 4

h −1

i 2

j −6

b 8

c 5

d −27

f −402

g 88

h −123

17 a −10

−5

−10

−5

Fluency 6 a 2

d −5

ii −2

15 a i 2

16 a −7

5 Integers that are the same distance from 0 on the number line, but are on different sides of 0.

c −8

18 16 Reasoning 19 False, as when the integer is positive, the opposite will be negative and hence less than the number itself.

Answers mathspace.co

793

20 0 serves as the centre point of the number line, and hence the starting point of the positive and negative integers. 21 Integers are opposites if their locations on the number line are the same distance from 0. If we start at 0 we also end up at 0. 22 8 is an integer value as it is a whole number. If north is considered positive, then 8 km due north can be represented by 8 and 8 km due south can be represented by −8. The integers 8 and −8 are opposite. 23 Some possible answers: Early civilisations believed only countable numbers were real. There was no need for place holders as numbers were not large. 24 No, infinity is not an integer. It is not a number that can be expressed as a fixed value on the number line. Infinity is a concept used to represent a quantity that is unbounded or limitless.

Fluency 6 a i −3, 9, 14

ii 14

b i −18, −3, 0

ii 0

7 a i 1, −4, −18

ii −18

b i 10, 4, −1

ii −1

8 a 13

b −7

c 8

d 35

9 a −21

b −24

c −1

d −36

10 a 8 < 10

b 2 > −4

c −5 < −1

d −3 > −8

11 a i Negative seven is less than negative two ii True b i Eight is greater than negative six ii True c i Zero is greater than negative one ii True d i T wenty three is less than negative forty seven ii False e i Seven is greater than and equal to four

Problem-solving

ii False

25 a i No

f i Three is less than and equal to two

ii Yes

b −400

ii False

26 a Arriving 14 minutes early

b −14

27 a Ascending 2 floors is represented by 2. b “A weight loss of 11 kg” is represented by −11. c “A temperature of 10 °C above freezing point” is represented by 10. 28 a 21

b −21

29 a −5100

b Being at sea level

b <

c >

d >

e >

f >

g <

h >

13 a −19 °C < 4 °C or 4 °C > −19 °C b −17 °C < −3 °C or −3 °C > −17 °C 14 a −15 < −2 or −2 > −15 b −10 < 0 or 0 > −10

3.02 Compare integers

Reasoning 15 Gian is correct as −5 is to the right of −7, therefore −5 is the greater integer.

Understanding 1 a 6

b −4

c 6

2 a −4

b −6

c −4

3 The number on the right hand side is the larger. 4 a greater than b less than c Order from smallest to largest d Order from largest to smallest 5 Ascending order

794

12 a >

Mathspace WA Curriculum – Year 7 mathspace.co

16 a $8000 profit b I would rather be $225 in the black because it means a profit, whereas $784 in the red means a loss. Profit is preferable to loss. 17 Krypton has the lowest boiling point. −153 °C is further to the left of zero than −62 °C. Problem-solving 18 −15 °C, −11 °C, −9 °C, 2 °C, 30 °C 19 −25 < −11 < 10 < 16 < 27

20 15 > 11 > 3 > −9 > −13 21 a 2, 4, 6, 8, 10, 12

b 2, 1, 0, −1, −2, −3

c −16, −12, −8, −4, 0, 4 d −4, −1, 2, 5, 8, 11 e 6, 3, 0, −3, −6, −9

f −17, −12, −7, −2, 3, 8

3.03 Add and subtract integers Understanding

14 0, because it is a neutral number and when you add or subtract zero you do not move left or right on the number line. 15 Yes, the result will be negative if the magnitude of the negative integer is larger than the magnitude of the positive integer. For example, −6 + 5 will be negative because the magnitude of −6 is larger than the magnitude of 5. 16 Negative. For example, −5 + −8 will be negative because we move further left on the number line.

1 Right 2 Left 3 Left

Problem-solving

4 Right 5 a Start with 3 and move 12 units left. b Start with −5 and move 9 units right. c Start with −2 and move 8 units left. d Start with −4 and move 11 units right.

17 a D

b 8 km

18 a 3

b −4

c −4

d 1 m below the surface

19 18 °C 20 a −6

Fluency

b 6m

6 a 14

b 5

c −3

d 0

21 185 m

e 19

f 7

g 4

h 7

22 525 metres

7 a 21

b 8

c −9

d 0

23 14 people

e −9

f 0

g 58

h −11

i 0

j −3

k 40

l 32

8 a 3

b −2

c 0

d −18

e 1

f 1

g −3

h −19

9 a 9

b −4

c 0

d −26

27 a i 254 mL

ii 243 mL

iii 250 mL

iv 251 mL

e −73

f −17

g −22

h 1

i −2

j 25

k −30

l 0

10 a 9

b −15

c 5

d 12

e 0

f 3

g 81

h 2

i 14

j −14

k −27

l −14

m −3

n 7

o −14

p 5

11 a 32

b 1

c 11

d 13

12 a False

b True

c False

d False

Reasoning 13 a i 0

ii 0

b The result is always 0 because you move the same distance left and right from zero.

24 −210° C 25 56 points 26 a 1

b −2

c 1

d 2

b 2499 mL 28 a i 5

ii −13

iii 11

b 3

Chapter 3 review 1 a Integer

b Non-integer

c Integer

d Integer

e Non-integer

f Non-integer

g Non-integer

h Integer

2 The number on the left hand side is the smaller. 3 a 3

b 5

c −2

d 0

4 a 7

b 11

c −6

d −6

e 3

f −2

5 a i −5, 11, 17

ii 17

Answers mathspace.co

795

b i −20, −1, 3

Fluency

ii 3

6 a −29

b −28

c −3

d −40

7 a <

b >

c >

d >

e >

f >

g <

h <

8 a 27

b −13

c −18

d 0

e 50

f −3

g 30

9 a 13

b −25

4 a

5 a

h 42

c −4

d −58

6 a i

f 0

g −28

h 0

b −21

c 0

d 14

b i

f −4

g −20

h 4

c i

b 1

c 16

d 19

d i

13 −20m, −13m, −8m, 5m, 25m

e i

14 −18 < −5 < 8 < 13 < 22

f i

e 23 10 a 18 e −20 11 a 49

12 a Descending 18 metres

15 a 3, 6, 9, 12, 15, 18 c −5, −2, 1, 4, 7, 10

b −18

b −14, −10, −6, −2, 2, 6 d 8, 5, 2, −1, −4, −7

16 17 °C

7 a

17 90 m 8 a

18 25 boxes 19 a i 506 g ii 495 g

iii 500 g

iv 503 g

b 5005 g

4.01 Identify, order and compare fractions

b 1

Understanding 1 a i 8

ii 9

b i 3

ii 7

c i 22

ii 50

d i 10

ii 3

2 a One half

c 2

b Three quarters

c Five hundredths

d Three and six eighteenths 3 a

d 1

d e 2

796

Mathspace WA Curriculum – Year 7 mathspace.co

Next, we need to ﬁnd the point that is halfway

between 1 and 2. Since each part represents

of the total distance from 0 to 2, we can count

two ticks to the right of 1 to reach the desired point.

9 a

10 a

11 a

f c True

d False

12 a False

b True

Therefore, the fraction represented by the point that is halfway between 1 and 2 on a number line from 0 to 2 divided into 8 equal parts is or . 17 First, we can draw two number lines of equal size and label them 0 and 1 on either end. Then, we can divide the line into equal parts to represent the denominators of the two fractions. In this case, we need to divide the line into 4 equal parts for

and 10 equal parts for

Reasoning 13 One half 14 One ﬁfth 15 To compare these fractions, we can convert them to have a common denominator. The smallest common denominator is 10, so we can write:

is greater than

, we know that

the greater fraction. Therefore,

be at a distance of

Then, we can see that the fraction

from 1 on the number line.

to the right of

To ﬁnd this point, we can start by identifying the

mark

that it is larger. Therefore, 18

is located

on the number line, meaning .

on the number line, and

on the

number line. 0

seven-tenths of the way from 0 to 1 on the number line.

equal parts, 1 would be located at the fourth tick

, we mark a point at

the seventh tick mark from 0, since

position of 1 on the number line. Since the distance between 0 and 2 is divided into 8

is three-quarters of the way from 0 to 1

on the number line. For

The point that is halfway between 1 and 2 would

we mark a point at the third tick mark from 0,

from 0 to 2.

line by locating their numerator values. For ,

16 If the number line from 0 to 2 is divided into 8 equal parts, each part would represent a distance of

Next, we can plot the fractions on the number

since Since

Visually, the midpoint between these points (equally distant) is

Answers mathspace.co

797

Problem-solving

d i

19 a

20 a

−1

−

3 4

−

1 2

−

1 4

1 2

3 4

ii 6 a

7 a

8 21 Suma 22 B

23 mango smoothie, apple smoothie, strawberry smoothie

10 a

4.02 Negative fractions

11 a

Understanding

Reasoning

1 Answer may look like:

12 Answer may look like:

The negative fractions have a negative sign (−) in front of them. Therefore, the negative

When we talk about direction in terms of distance, we use positive and negative numbers to represent movements in different directions. In this case, we are considering a person who is hiking both south and north. Let’s imagine that the starting point is 0 km.

and − .

fractions in the given options are 2 Each tick represents 3 Answer may look like: and

are located at the same distance from

km due south, they When the person hikes are moving in a negative direction (south) for a

zero on a number line, but on opposite sides.

distance of

km. This can be represented by

would be to the right of zero, while

the fraction

, which is a negative number.

would

On the other hand, when the person hikes

be to the left.

km due north, they are moving in a positive Fluency

direction (north) for a distance of

4 a

This can be represented by the fraction which is a positive number.

5 a i −1 − 6 − 5 − 4 − 3 − 2 − 1 0 7 7 7 7 7 7

1 7

2 7

3 7

4 7

5 7

6 7

−2

−1

798

and

would also be on

opposite sides of 0.

ii c i

Since positive and negative numbers are on opposite sides of 0 on a number line, the fractions

ii b i

km.

Mathspace WA Curriculum – Year 7 mathspace.co

In summary, the fractions representing hiking due north and due south are on opposite sides of 0 because they represent movements in opposite directions, with one being positive and the other being negative. 13 Answers may look like: We can use a number line to compare negative fractions.

−1

−

8 9

−1

−

7 9

5 6

−

and

2 3

−

5 9

2 3

−

4 9

−

1 2

−

1 3

−

2 9

1 3

−

1 9

Understanding

1 6

are located at the same spot on the

number line because they are equivalent fractions, meaning they represent the same value. As we can see from the number line, closest fraction to zero.

is the

3 − 4

−1

−

5 6

−

5 − 8

1 − 2

3 − 8

2 3

−

1 2

−

1 − 8

1 6

1 − 4

1 3

−

We can use the number lines to arrange the fractions from smallest (furthest from 0) to largest (closest to 0). In ascending order:

Problem-solving 15 Answers may look like: 1 8 −1

−

7 8

1 8

−

3 4

1 8

−

1 8

5 8

−

1 2

Since we have moved from we have moved right

3 8

−

1 4

200 m

1 8

(positive fraction).

1 of 600 = 200 m 3

5 C, D Fluency 6 a

7 a

8 a

9 a Yes

b No

c No

d Yes

b and

11 a

12 a

13 a i 35

400 m 600 m 800 m

−

and

16 Answers may look like: 0m

Write both fractions with a common denominator first.

10 a

1 8

−

2 Answer may look like: Simply compare their numerators. The fraction with the larger numerator is the larger fraction.

We can use a number line to compare negative fractions. 7 − 8

3 Answer may look like:

14 Answers may look like:

−1

4.03 Equivalent and simplified fractions

ii +

1000 m

1000 m below sea level

2 3

iii b i 32 ii

Answers mathspace.co

799

We can say the same for . Therefore,

iii

and

are not equivalent fractions.

14 a i 55

For , we can simplify by dividing both the

numerator and denominator by 3, giving us . For

iii

, we can simplify by dividing both the

numerator and denominator by 4, giving us . b i 65

Therefore,

and

are equivalent fractions.

In conclusion, the pairs of fractions that are equivalent are

iii

and , and,

and

18 Answers may look like: Reasoning 15 Answers may look like: Yes,

and

are equivalent fractions. To see

why, we can simplify each fraction to lowest terms.

is already in lowest terms, while

Yes a fraction with prime numbers in numerator and denominator is in fully simplified form as there is no common factor that the numerator and denominator can be divided by, other than 1. Problem-solving

can be simpliﬁed by dividing both the

numerator and denominator by 4 to get .

Therefore, the two fractions are equivalent.

8 4

16 Answers may look like: To ﬁnd an equivalent fraction to

with a

denominator of 12, we need to multiply both the numerator and denominator by the same factor. We can start by multiplying the denominator of , which is 3, by 4 to get 12. To keep the fraction equivalent, we also need to multiply the numerator by 4. Therefore, an equivalent fraction to

with a denominator of 12 is

17 Answers may look like: To determine which pairs are equivalent, we need to simplify each fraction to its lowest terms. For , the only factor that can be divided out is 1, so it is already in lowest terms. For , we can simplify by dividing both the numerator and denominator by 2, giving us . Therefore,

and

are equivalent fractions.

For , the only factor that can be divided out is 1, so it is already in lowest terms.

800

Mathspace WA Curriculum – Year 7 mathspace.co

Blue

Green

Red

Did not answer

Example answer: 4 students like blue, 4 students like green, and 8 students like red. 8 students did not answer. 20 Answers may look like: If 3 out of 8 pizza slices have been eaten, then 5 slices remain. Therefore, the fraction of the pizza that remains uneaten is . An equivalent fraction for the remaining fraction with a denominator of 16 is

21 Answers may look like: has the simpliﬁed fraction

(divide the

numerator and denominator by 5).

Multiply both numerator and denominator by 7

will like blue in the class next door

4.04 Add and subtract fractions

roup the whole number G parts and fraction parts

Add the whole number parts and fraction parts

Understanding is correct.

(equivalent fraction). So, 7 students

to have

2 Answer may look like:

Evaluate the sum

10 a

Simply add or subtract the numerators and keep the same denominator. 3 Answer may look like: Write both fractions with the same denominator. 4 Answer may look like: Convert the mixed numbers to improper fractions, write with a common denominator and then add as usual.

11 a 7

c Rewrite as improper fractions

Fluency 5 a 6

b 36

c 20

d 12

6 a

Rewrite with a common

7 a b

13 a

Evaluate the sum

Rewrite as a mixed number

ewrite with a common R denominator

Evaluate the difference

Rewrite with a common denominator

Evaluate the difference

Rewrite as a mixed number

Evaluate the difference

b Example answer:

8 a

denominator

ewrite with a common R denominator

Rewrite with a common denominator

14 a

15 a

Reasoning 16 a

b c Both methods are equally efficient however the second method involves a negative fraction which is more likely to cause errors.

Answers mathspace.co

801

17 Answers may look like: To solve this problem, we need to ﬁnd the total fraction of the pizza that Teddy and Archie ate and subtract it from 1 (the whole pizza). Teddy ate

of the pizza, and he gave

12 cm2 is

of the cake. 9 cm 1 6

of the

pizza to Archie, so the total fraction of the pizza

8 cm

they ate is:

4 cm

3 cm

To ﬁnd how much of the pizza is left, we subtract

After cutting this piece, what was left of the

from 1:

cake is:

Therefore,

of the original pizza is left.

We can also check our answer by adding the fractions that Teddy and Archie ate and seeing if it equals the fraction that is left:

Rewrite with a common denominator

Evaluate the difference.

of the cake will be left.

So, our answer is reasonable.

Problem-solving

18 Answers may look like:

To complete the recipe, Wei needs to have cup of sugar, but she only has

cup of sugar.

To ﬁnd out how much more sugar Wei needs, we need to subtract

from :

Therefore, Wei needs complete the recipe.

cup more sugar to

This answer is reasonable because cup is less than half a cup, which is the amount of sugar Wei already has. It makes sense that she would need less than what she already has in order to complete the recipe. 19 Answers may look like: To determine the fraction of the cake that is left after cutting a piece, we need to compare the area of the piece that was cut to the area of the original cake. The original cake measures 9 cm by 8 cm, so its area is: 9 cm × 8 cm = 72 cm2 The piece that was cut measures 4 cm by 3 cm, so its area is: 4 cm × 3 cm = 12 cm2

802

Mathspace WA Curriculum – Year 7 mathspace.co

hours

cups

gram

4.05 Fractions of a quantity Understanding 1 Answers may look like: Yes,

is equivalent to

This is

because of the commutative property of multiplication, which states that the order of factors does not affect the result of a multiplication operation. 2 Answers may look like: Let’s say we want to ﬁnd Using a diagram:

of 50.

The diagram below represents the total quantity of 50

i 80

j 14

k 40

6 a 390 L

b 200 m

c 1100 kg d 414 m2

e 135 m

f 165 kg

g 225 L

7 a 20 cents

Divide the rectangle into 5 equal parts, since the denominator of the fraction is 5,

l 180 h 380 cm

b 72 seconds

c 16 days

d 5400 mL

e 42 g

f 36 mm

g 100 cm

h 9000 mg

i 24 minutes

j 7200 m

k 1150 cents

l 1.05 L

m 1320 minutes

n 1980 hours

o 336 days

p 1.1 kg

Reasoning 8 a Then we can shade in 2 of those equal parts to represent the

of the quantity.

b 360

c Georgia

d In part (a), we know the total and the quantity handed out, so we are finding the fraction using the two quantities. In part (b), we know the fraction handed out so we have to find the fraction of the total to find the quantity handed out. They are effectively the reverse of each other, in terms of working. 9 Answers may look like:

Counting the number of shaded parts and multiplying by 10, we can ﬁnd that

of 50 is:

Multiplying fractions: To ﬁnd of 50 using multiplication, we can multiply the fraction by the quantity:

There are 180 students in the drama club. To explain our reasoning, we used multiplication to ﬁnd

of the total number of students in the

school. We then simpliﬁed the resulting fraction to get the ﬁnal answer of 180 students in the drama club. 10 Answers may look like: If of the marbles in the jar are blue, then the remaining fraction of marbles that are red is

Therefore, we have found that

of 50 is 20, both

by using a diagram and by multiplying fractions. 3

Fluency 4 a 7 e 56 5 a e 10

To ﬁnd how many red marbles are in the jar, we can multiply the total number of marbles by the fraction that are red:

So there are 300 red marbles in the jar. b 32

c 3

d 27

h 40

If not done previously, to ﬁnd the fraction of marbles that are red, we can divide the number of red marbles by the total number of marbles:

d 90

f 7

g 24

h 45

Therefore,

of the marbles in the jar are red

Answers mathspace.co

803

11 Answers may look like:

2 Answer may look like:

The row with the error is the second row. The correct working is below:

is equivalent to 1, so multiplying it with another fraction will result in the same fraction that we started with. This is because any number multiplied by 1 is equal to itself.

Number of apples

Number of pears

Number of oranges

Therefore, there is only 1 orange in the fruit bowl. Problem-solving

3 a Yes

b No

c Yes

d No

4 a

b 7

h 1

5 a

6 a

7 a 1

Fluency

12 $4 13 270 pages 14 12 15 Sister has $16 000 so he would have $24 000

e 1

16 $320 17 a A fair coin should land on tails 50% of the time. That is equivalent to

or . This coin is

landing on tails a bit more than expected, which is . b 25 c 20 times assuming the coin is fair, we can expect that it has an equal chance of landing on heads or tails on any given ﬂip.

18 Fe scored 35 points, Jamie scored 22 points, Sarah scored 20 points

19 a 1159 L is used in total

b 551 L

4.06 Multiply fractions

11 The farmer will use planting.

Understanding 1 a To multiply

and

using shading:

We can draw two rectangles, one with four equal parts and one with three equal parts, and shade in one part of the first rectangle and two parts of the second rectangle. Then, we can combine the two rectangles and count the number of shaded parts in the combined rectangle to get the product. b To multiply

and

using multiplication:

You multiply the two numerators together and the two denominators together. The resulting product is your answer.

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Mathspace WA Curriculum – Year 7 mathspace.co

years

cups

of an acre of the land for

of an acre remains.

Reasoning 15 Answers may look like: If the recipe calls for

cup of sugar, and you

only want to make one-third of the recipe, then you need to find one-third of

cup.

To find one-third of by

cup, we need to multiply

Problem-solving 18

So, you only need

cup of sugar to make

one-third of the recipe.

cups of sugar is needed to make 5 cakes

16 Answers may look like: Cynthia is incorrect because if each juice box contains

c $2.50 ÷ 10 = $0.25 per cup

boxes will contain 6 times that amount of juice. cup by 6 to find the total

amount of juice in the pack. Amount of juice in the pack

cup × 6

boxes = 4 cups So, there will be 4 cups of juice in the pack. To find the area of the rectangular garden, we need to multiply its length by its width. metres, which

can be written as an improper fraction:

The width of the garden is

$0.25 per cup ×

cups = $0.625

It would cost approximately $0.63 to buy the sugar needed to make 5 cakes.

4.07 Divide fractions Understanding 1 Answer may look like:

17 Answers may look like:

The length of the garden is

cups

cup of juice, then a pack of 6 juice

We can multiply

cups

20 a

metres, which

can also be written as an improper fraction:

Now we can multiply the length and width to find the area:

Dividing fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 2 a

3 4 Answer may look like: Yes, you can divide a fraction by itself, and the result is always equal to 1. 5 a

We can convert the improper fraction to mixed fraction. Fluency Therefore, the area of the rectangular garden is square metres. 10 square metres of turf will

6 a 33

cover it.

Answers mathspace.co

805

7 a

18 Answers may look like:

e 1

h 7

8 a

would be half this time - divide

of an hour. To read the whole book, we would

If it takes

of an hour to read

of a book, we

can use reasoning and proportion to ﬁnd out how long it will take to read the entire book. If it takes

of an hour to read , then to read 1 third by 2 to get

take 3 times this long, which is hour.

9 8 days

of an

10 16 glasses 11

Therefore, Sam is correct.

12 30 days

Problem-solving

13 5 presents

19 The tank needs to be ﬁlled with an additional

14 7 loaves

412.5 litres of water to reach

15 9 bows

20 a i 12 letters ii 9 letters b

Reasoning 16 Answers may look like: To divide

by , you need to invert the second

fraction and multiply it with the ﬁrst fraction. So, the correct fraction to invert is . When you invert a fraction, you ﬂip it upside down. So, inverting

gives you

or simply 4. Then,

you can multiply the ﬁrst fraction

by the

inverted second fraction (4) as follows:

cm and

of its capacity. iii 6 letters

cm wide

4.08 Order of operations with fractions Understanding 1 • Evaluate expressions inside the brackets • Evaluate multiplication or division from left to right • Evaluate addition or subtraction from left to right 2 Answers may look like: The brackets in the expression

Therefore,

divided by

mean that instead of doing the

is equal to .

division first, we perform the subtraction inside

17 Answers may look like:

the brackets first, before dividing

We would divide the 8 slices by :

result. 3 a Multiply

So, we could feed 10 people (not 11 as whole person). We can check this as:

by the

and

b Divide

c Divide

d Multiply

and

e Add

and

f Multiply

and

h Multiply

and

is not a

This is just under the 8 slices. g Subtract

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Mathspace WA Curriculum – Year 7 mathspace.co

from

12 Answers may look like:

Fluency

To solve the problem, we need to ﬁnd the total number of students going on the trail walk, which is the sum of the students in the ﬁrst scout group and the students from the second scout group.

4 a

5 a

he ﬁrst calculates the number of students from

the ﬁrst scout group who are going on the trail

6 a

, which means

Riley’s expression is

walk

and then adds the 20 students

from the second scout group. Riley’s calculation for the ﬁrst part of the

7 a

, which means 9

expression is

students from the ﬁrst scout group are going on the trail walk. Adding the 20 students from the second scout group gives us a total of 29 students on the trail walk. , which means

Reese’s expression is

he ﬁrst adds the number of students in both 8 a

9 a

c False

d True

f 15

10 a True

b False

e False

f False

Reasoning 11 Answers may look like: To simplify the expression

, we must

follow the order of operations, which is a set of rules that tells us which operation to perform ﬁrst when there are multiple operations in an expression. The order of operations is as follows: • Perform any calculations inside parentheses ﬁrst. • Multiply or divide from left to right. • Add or subtract from left to right.

scout groups (12 + 20) and then multiplies by . Reece’s calculation for the first part of the expression is (12 + 20) = 32, which means there are 32 students in total. Multiplying 32 by gives us 24 students going on the trail walk. So, Riley and Reese do not get the same result. Riley gets 29 and Reese gets 24. This is because they are calculating different things. Riley is finding the number of students from the first scout group going on the trail walk and adding the students from the second scout group. Reese is finding the total number of students going on the trail walk, including both scout groups. 13 Answers may look like: To find the area of the composite shape, we need to add the areas of the two rectangles together. Ruby uses the expression to find the area, which simplifies to (9 + 6.5) = 15.5 square units. Jade, on the other hand, uses the expression

Using the order of operations, we can simplify the expression

to ﬁnd the area.

by multiplying ﬁrst. This simpliﬁes to

square units.

Answers mathspace.co

807

So, both Ruby and Jade get the same result for the total area of the composite shape, which is 15.5 square units. Therefore, both methods are correct and will give the same answer.

8 a

b Yes

c No

d Yes

However, Jade’s method involves ﬁnding the area of the composite shape as the difference between the area of a larger rectangle and the area of a smaller rectangle, whereas Ruby’s method involves adding the areas of two separate rectangles.

10 a

Problem-solving

11 a i 12

14 720 people were surveyed

hours, or 1 hour and 24 minutes.

9 a Yes

iii b i 70

Chapter 4 review

ii 1 a

d iii

2 a i

b i

3 a

12 a

13 a 7

b 1

c 8

f 1

d 14

4 a 0

b 7

16 a

c 5

17 15 cups

d 9

5 a False 6 a i

b False

c True

−1

d True

18 a 1

ii b i

−2

−1

20 8 cakes

21 7 a

Mathspace WA Curriculum – Year 7 mathspace.co

kilometres

22 7 batches 23 a

808

servings

5.01 Decimals and place value

15 Just because a number has more digits after the decimal point doesn’t always mean it’s larger. To compare decimals properly, look at the value of each digit based on its place.

Understanding 1 10 tenths 2 a 100 hundredths 3 a iii

b 10 hundredths

b i

c ii

Fluency 4 a 7 hundredths c 7 ones

Problem-solving

5 a 0.71

b 0.4

c 1.605

d 4.9999

6 a 8

b 2

c 9

d 5

c 0.39 > 0.389

For example, in 0.8, the digit in the tenths place is 8, while in 0.672 33, the digit in the tenths place is 6. Since 8 is bigger than 6, we can say that 0.8 is larger than 0.672 33, even though it has fewer digits after the decimal point.

b 7 thousandths d 7 tenths

7 a 2.305 < 2.309

values, shifting the decimal point left or right. This shifting does not occur when multiplying or dividing by other numbers, which requires more complex arithmetic.

b 20.9999 > 20.9998 d 0.005 < 0.01

16 a Tuesday b Wednesday and Friday 17 a Daniel, Dylan, Nicky, Bernice b 2nd 18 9.633

e 0.483 > 0.4 8 a 0.0024, 0.024, 0.204, 0.24, 2.4

5.02 Add and subtract decimals

b 3.068, 3.1, 3.68, 3.681, 3.86 Understanding

9 a 0.54, 0.504, 0.5, 0.054, 0.0054

1 Answer may look like:

b 7.82, 7.8, 7.28, 7.168, 7.078 10 a 5.6

b 52

c 35

d 674

e 648.9

f 5761

g 3.784

h 0.03

i 930

j 5943

k 40

l 8

11 a 3.4

b 0.0832

c 0.675

d 0.048

e 9.535

f 75.835

g 0.057

h 0.39

i 0.7966

j 0.423

k 0.4233

l 0.0022

Reasoning 12 Since both digits in the ones place are the same, we compare the digits in the tenths place. In 3.45, the tenths digit is 4. In 3.045, the tenths digit is 0. Since 0 is smaller than 4, we can conclude that 3.045 is smaller than 3.45. 13 a Right b Two places because there are two zeros in 100. c 97.3 14 Multiplying and dividing decimal numbers by powers of 10 is different from other numbers because powers of 10 uniquely affect place

When adding decimals we have to make sure that the decimal points are lined up. 2 Answers may look like: The decimal points are not lined up. Fluency 3 a 0.4

b 8.4

c 13.5

d 0.71

e 0.56

f 13.91

g 0.56

h 5.7

b 14.248

c 60.38

d 148.11

4 a 4 e 168.21 5 a 0.65 e 6.723

f 327.992 g 937.444 h 844.682 b 1.72

c 1.856

d 23.91

f 89.142

6 17.865 7 $33.83 8 18.445 9 109.65 10 a 1.9

b 4.2

c 3.1

d 5.7

e 1.3

f 0.22

g 0.24

h 2.5

Answers mathspace.co

809

11 a 4.29

b 56.43

c 7.89

d 0.88

e 11.04

f 35.88

g 0.356

h 9.129

9 9 - 6 4

. .

9 10 9 5

12 93.52 13 20.43

In the hundredths column, 0 is less than 5, so we need to take away 1 from the tenths, and carry it to the ones which gives us 10. Since the tenths, units and tens digits are all zeros, the taking away goes up to the hundreds place.

14 $68.89 15 75.913 16 $1.80 Reasoning 17 a

1.2

0.5

1.6

1.5

1.1

0.7

0.6

1.7

1.0

2.4

1.8

4.8

5.4

3.0

0.6

1.2

4.2

3.6

2.4

2.8

0.8

0.4

2.0

3.6

3.2

1.2

1.6

9 9 - 6 4

9 10 9 5 5

. .

9 10 9 5 0 5

Subtract the tenths, 9 − 9 = 0: 5

0.9

0.6

3.9

4.8

4.2

4.5

1.5

1.8

2.1

3.6

2.7

3.3

3.0

2.7

1.2

9 9 - 6 4 5

. . .

9 10 9 5 0 5

Bring down the decimal point and subtract the units, 9 − 4 = 5: 6

9 9 - 6 4 3 5

. . .

9 10 9 5 0 5

100 − 64.95 = 35.05 Subtract the tens, 9 − 6 = 3:

8.23, 6.45 and 7.86

The change is $35.05.

9.12, 4.63 and 8.79 19 The row that started the error is row 2. 1

. .

So now we have 10 − 5 = 5:

18 Answer may look like:

9 9 - 6 4

0 0 6 4

. .

0 0 9 5

Align the decimal points.

Problem-solving 20 $21.95 21 0.65 seconds 22 Group 3 23 0.54 kg 24 a The processes from lowest to highest defect rate: • Process C has a 0.0225 defect rate • Process A has a 0.023 defect rate • Process D has a 0.025 defect rate • Process B has a 0.032 defect rate b The smallest difference is between Process C and Process A with a difference of 0.0005.

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Mathspace WA Curriculum – Year 7 mathspace.co

5.03 Convert between fractions and decimals Understanding 1

Ones

Tenths

Hundredths

2 Convert the denominator to a power of 10 such as 10, 100, 1000. 3 To convert decimals to fractions, we can start by counting the number of decimal places, that is, the number of digits after the decimal point. 4 If there are 2 decimal places then our fraction will have a denominator of 100. Fluency

12 a 0.08

b 0.375

c 0.156

d 0.313

e 0.175

f 0.163

g 0.219

h 0.225

13 a 2.75

b 5.5

c 1.875

d 4.4

e 3.3

f 6.833

g 2.714

h 8.667

14 a

e f 15 a

5 0.53 6 a

b 0.7

d 0.3

7 a

b 0.67

d 0.33

8 a 0.1

b 0.2

c 0.7

d 0.01

e 0.27

f 0.93

g 0.001

h 0.006

i 0.068

j 0.111

k 0.747

l 0.954

m 1.3

n 9.4

o 6.98

p 5.527

19 3.75 litres

9 a 1.1

b 9.3

c 9.65

d 7.12

Reasoning

10 a

0.01

0.001

0.5

0.25

0.75

0.2

0.05

0.02

18 0.375 cups

b False, because the unshaded portion is less than half of the total rectangle. c True. From the image, it appears that the shaded portion is more than 0.6 but less than 0.8. Yes, a fraction can have a decimal equivalent that is greater than 1 if the numerator is greater than the denominator. 22 a

Fraction Decimal

17 a 0.28

21 Answer may look like:

Fraction Decimal

0.1

Fraction Decimal

20 a True, because the shaded portion is larger than the unshaded portion.

Fraction Decimal

16 a 0.8

Fraction 0.4

0.3

0.14

11 a 0.24

b 0.4

c 0.46

d 0.76

e 0.35

f 0.73

g 0.716

h 2.25

i 1.5

j 1.25

k 1.24

l 8.68

Decimal

0.5 0.25 0.20 0.125

0.10

b Answer may look like: As the denominator of a fraction increases, the decimal value of the fraction becomes

Answers mathspace.co

811

smaller. This is because the larger denominator means that the whole is divided into more parts, resulting in smaller individual parts and a smaller decimal value when the fraction is converted to a decimal.

Fluency 4 a i 2

ii 0.15

b i 3

ii 0.108

c i 3

ii 0.006

d i 3

ii 0.040 = 0.04

e i 4

ii 0.0015

f i 6

ii 0.000 042

g i 4

ii 0.0090 = 0.009

h i 4

ii 0.0030 = 0.003

i i 3

ii 0.036

j i 3

ii 0.064

k i 2

ii 1.26

24 a

l i 3

ii 0.385

5 a 20

b 200

c 2

d 0.2

e 0.18

f 8.8

g 3.84

h 8.46

i 38.4

j 595.2

k 27.23

l 19.04

6 a False

b False

c True

d True

Problem-solving 23 a b c

c 25 0.945 hectares 26 The third friend paid $45. 27 The baker needs 3 cups of flour, 1.5 cups of sugar and

cup of milk.

7 a 2 0.3 0.04 5

Understanding 1 Answer may look like: When you multiply two decimals that are less than 1, the product will be smaller than the decimals being multiplied. For example, 0.5 × 0.3 = 0.15, which is smaller than both 0.5 and 0.3. 2 a Greater than both numbers. b In between the two numbers. c Less than both numbers. d Greater than both numbers.

1.5

0.20

8 a 40 6 0.2 6

5.04 Multiply decimals

240

1.2

9 a 4

0.9

0.02

0.6

0.54

0.012

2.4

b 11.70

10 a 2 0.3 3

0.9

0.1

0.2

0.03

11 a 0.8 0.07 7

5.6

0.49

0.8

0.64

0.056

b 277.2

b 2.952

b 7.13

b 6.786

12 0.4183

e In between the two numbers.

13 4.1796

f Less than both numbers.

14 $8904

g Greater than both numbers. h In between the two numbers. i In between the two numbers. 3 a When only one of the numbers is less than 1. b When both numbers are less than 1.

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Mathspace WA Curriculum – Year 7 mathspace.co

Reasoning 15 Answers may look like: To get the answer to 3.6 × 4.2, we can again multiply the two numbers: 3.6 × 4.2 = 15.12. This is because we can ignore the decimal points and treat the numbers as if they were

whole numbers, then count the total number of decimal places in the factors, which is 2 in this case and place the decimal point that many places from the right in the product. 16 Answers may look like: 12.588 84 because the ﬁrst decimal number has 3 decimal places while the other has 2, for a total of 5 decimal places. 17 a $7.00

b 0.3

4 a A 5 a 79.38 ÷ 4.2 = 18.9 4.2 × 18.9 = 79.38 b 4.5 = 18.9 ÷ 4.2 4.2 × 4.5 = 18.9 c 141.12 ÷ 5.6 = 25.2 25.2 × 5.6 = 141.12 d 5.6 = 25.2 ÷ 4.5 5.6 × 4.5 = 25.2

b Answers may look like: The revenue earned by selling a product is calculated by multiplying the price per unit by the number of units sold. If each loaf of bread costs $3.50, and the bakery sells 100 loaves in a week, they make 100 × $3.50 = $350. c If the price of each loaf of bread is increased by $0.25, the new price per loaf would be $3.75. Selling the same amount of loaves, which is 100 at this price would result in a revenue of 100 × $3.75 = $375. This is an increase of $25 from the original revenue of $350. Problem-solving 18 a 0.4 cm2

Fluency 6 a i 10

iii 0.2

7 a 8.08

b 5.77

c 8.78

d 81.79

e 56.24

f 65.4

g 89.72

h 132.65

b 33.6 ÷ 3

8 a 10 c 11.2

b 480.8 ÷ 4

9 a 100 c 120.2 10 a 80

b 600

c 20

d 104.1

e 4000

f 4

g 9

h 42

i 1401

j 900

k 3000

l 800

b False

c True

d False

11 21 days.

b 0.014 m2

19 2.25 cups of sugar and 1.5 cups of butter.

12 17 13 a True

20 16.6 km 21 $365.57

Reasoning

22 $108.71

14 a 235 d 235 000

23 $26.86

b 42

c 23.5

e 0.002 35

f 42

15 Answer may look like:

24 100 kg 25 There are 98 small boxes, while the total weight of all the boxes is 157.5 kg. 26 5880 pieces

Understanding 1 a 68.4 ÷ 6

b 11

2 a A

b A b 4

The expression 1.6 ÷ 0.4 can be simpliﬁed as follows: Jo multiplies both numbers by 10 before dividing. So, the expression he evaluates is:

5.05 Divide decimals

3 a 490

ii 2

b 12.2

c 99

Isla multiplies both numbers by 1000 before dividing. So, the expression she evaluates is:

d 8

Both Jo and Isla get the same answer, which is 4. This is because when you multiply both the

Answers mathspace.co

813

numerator and denominator by the same non-zero number, you get an equivalent fraction. Therefore, the value of the fraction does not change. In this case, Jo and Isla multiplied both the numerator and denominator by different numbers, but the result is still the same, because they multiplied by equivalent numbers (10 and 1000, respectively). Therefore, they will get the same answer, which is 4. 16 a i 5 cm ii He has increased the area which would have the effect of increasing the width and also decreased the divisor which will have the same effect. b i 5 cm ii He has decreased the area which would have the effect of decreasing the width and also increased the divisor which will have the same effect. c i 5 cm ii The changes he has made have opposite effects. Increasing the area would tend to increase the width, but he has also increased the divisor which would tend to decrease the width. Problem-solving 17 Juliet got better value, paying approximately $3.91 per kilogram. Luna’s bananas were $4.04 per kilogram. 18 Oliver will need 33 complete bottles and a third of the 34th. 19 160 cups 20 a i 7.5 m

4 a C

b 6

5 a B

b 25

Fluency 6 a 3

b 6

c 24

d 34

e 1

f 0

g 1

h 90

7 a 0.1

b 7.4

c 3.6

d 9.0

e 27.9

f 289.4

g 0.7

h 10.0

8 a 2.90

b 125.46

c 14.29

d 67.00

e 0.01

f 13.00

g 100.00

h 131.10

9 a 0.4

b 8.4

c 19.4

d 0.3

e 3.6

f 20.0

g 13.9

h 1.0

10 a 2.70

b 393.59

c 0.44

d 7.36

e 80.00

f 2.00

g 149.96

h 0.10

11 a 11.848

b 2.160

c 4.504

d 1.200

e 90.034

f 0.599

g 41.501

h 10.000

12 a 5.6596

b 5.6010

c 12.0067 d 40.0000

13 a C

b Lower

c 108 km

14 a 70 pages

Reasoning 15 Opie needs to calculate the decimal value up to four decimal places. The number on the fourth decimal place is the basis whether to round up or round down the number in the third decimal place. 16 a She should only use the 4th decimal place to round to three decimal places. b i 0.321 5

ii 10.9 m

b i 20 frames with no metal left over. ii 13 frames with 8.3 m metal left over. iii 8 of each type of frame with 2.8 m metal left over.

5.06 Round decimals Understanding

b Go back to the original number. 4.449 is closer to 4.4 than 4.5, so should round to 4.4. Once rounded to two decimal places 4.45 rounds to 4.5, which is further from the original number. 18 It is a convention (an agreed upon standard) and is consistent with the check digit being 5. Problem-solving

2 39

19 a 30 cm

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Mathspace WA Curriculum – Year 7 mathspace.co

ii 0.322

17 a 4.45

1 11

3 B and D

b Lower

c 71 pages

b Overestimate, both length and width are rounded up. c 28.88 cm 20 a $50

d 1.1 cm

16 $88.45

b Yes.

17 1.8 cups of cocoa powder and 1.2 cups of milk.

c No, because the exact total cost is $50.60 d No, by doing this Tommy has estimated the cost as lower than it actually is. 21 a 12 g and 16 g c Overestimate

b 4g

4 a 1.8 e 5.33

c 5

c No

6.01 Fractions, decimals and percentages d 1000 Understanding

f 79.465 b 3.1

c 6.7

d 0.23

1 a Multiply

f 6.87

g 0.427

h 53.21

2 100

5 $5.50 6 a 0.17

b Yes

d No, by doing this Ally has estimated the cost as lower than it actually is.

2 3.7, 0.37, 0.073, 0.0073, 0.00 073 e 1.71

19 Sally will need 42 complete containers and two-thirds of the 43rd.

21 a $420

1 1.351

b 0.74

18 $7.00

20 120 packets

d 3.4 g

Chapter 5 review

3 a 0.7

b Divide

3 50% b 0.32

c 0.54

d 0.45

e 0.768

f 1.75

g 1.28

h 0.25

i 3.74

j 6.75

7 a

4 45% 5 a b c d

8 0.84 cm 9 a 0.28

b 0.0012

c 0.0054

d 0.040 = 0.04

e 3.96

f 13.05

g 622.5

h 19.26

Fluency 7 a i 25%, 50%, 75%, 100%

10 a 9.09

b 5.08

c 9.76

d 80.12

e 60.18

f 80.1

g 96.55

h 170.1

b 4

c 600

d 102.5

f 900

g 3000

h 800

11 a 70 e 4000 12 22 days. 13 a 3.2457 c 10.0569 14 1.18 kg 15 a

6 The value is greater than 1 whole, or more than 100 out of 100.

ii 0.25, 0.5, 0.75, 1.0 b i ii 0.33..., 0.66..., 1.0 c i 20%, 40%, 60%, 80%, 100% ii 0.2, 0.4, 0.6, 0.8, 1.0

b 7.8313 d 22.9995

d i 1 2.5%, 25%, 37.5%, 50%, 62.5%, 75%, 87.5%, 100% ii 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1.0 e i 1 0%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100% ii 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0

Answers mathspace.co

815

8 a 0.18

b 0.57

c 0.234

d 0.055

e 1.2

f 3.03

g 5.00

h 68.97

9 a 7%

b 85.7%

c 530%

d 2400%

f 4.1%

g 0.35%

h 137.5% d 80%

e 1304% 10 a 47%

b 68%

c 55%

e 16.67%

f 61.11%

g 63.89% h 53.33%

i 650%

j 235%

k 720%

l 332%

11 a 20%

b 25%

c 62%

d 25%

12 a 0.485

b 0.082

c 0.068

d 0.026

e 0.043

f 0.135

g 0.222

h 0.618

13 a

14 a

c For example: Can get a more than 100% pay raise. 19 20 a C

b A

c D

d B

d 46%

21 For example:

22 a

b 100

23 No. A 180% increase means customers are now 280% of the original (100% + 180% = 280% = 2.8 times the original). Tripling would be a 200% increase (300% of original = 3 times). So, a 180% increase is less than tripling. Problem-solving

Decimal

Percentage

0.7

70%

0.53

53%

0.02

0.006

0.6%

1.23

123%

4.3

430%

2.8

280%

11.25

1125%

Fraction

24 15% 25 70% 26 a

b 0.2

c 20%

27 203 marbles are not red 28 Test 1, 64%. Test 2, 67.5%. Juliet did better on Test 2. 29 a 80%

b 60%

30 270% 31 $65 32 The recreational area is 27% while the greenhouse is 10%.

6.02 Compare percentages Understanding

16 a 450%

b 367%

c 904%

d 1382%

e 225%

f 220%

g 750%

h 990%

1 a

b 50% b 40%

17 a 77.7778%

b 57.1429%

2 a

c 91.6667%

d 95.2381%

3 Convert all three numbers to the same format in order to compare them. So all percentages, or all fractions or all decimals.

Reasoning 18 a Yes b No. Test score maximum is 100%. You can’t score more than 100% in the test.

816

Mathspace WA Curriculum – Year 7 mathspace.co

Fluency 4 a i 50%, 100%, 30%

b i 20%, 70%, 35%

6.03 Express a quantity as a percentage

Understanding

c i 37.5%, 20%, 80% ii d i 20%, 45%, 27% 5 a 13%

1 a 30% is the same as 30 out of 100 and 3 out of 10.

b 15% is the same as 15 out of 100 and 3 out of 20.

b Same

c Same

6 a 69% > 0.65

b 71% > 0.59

c 51% < 0.87

d 0.052 > 4.9%

c 75% is the same as 75 out of 100 and 3 out of 4. d 80% is the same as 80 out of 100 and 4 out of 5. e 50% is the same as 50 out of 100 and 1 out of 2. f 6 2% is the same as 62 out of 100 and 31 out of 50.

7 a True

b True

c True

d True

e True

f False

g True

h False

2 C

8 a False

b True

c False

d False

Fluency 3 a i

b i

c i

d i

11 For example: 0.6930, 0.6935, 0.6937

e i

12 For example:

f i

13 For example:

g i

h i

9 a c 92%, 60.1%, 0.36, 0.1 d Reasoning

10 For example: 88%, 88.5%, 89%

Problem-solving 14 a

4 a i

ii 42%

b i

ii 80%

c i

ii 5%

d i

ii 50%

e i

ii 175%

15 40

f i

ii 40%

16 Shop C

g i

ii 160%

b i > c

ii <

iii >

iv <

17 Stella

Answers mathspace.co

817

5 a i 40%

4 It means that the percentage is more than the original quantity.

ii 12%

h i

iv 20%

iii

Fluency

b i 75%

ii 35%

5 a i 3

c i 220%

ii 350%

b i 14

ii 3.5

iii 49

c i 7

ii 9

iii 63

d i 7

ii 3

iii 21

6 a 18%

d 25%

7 a 20%

b 54%

e 10%

ii 7

6 a 0.9 × 20 = 18 c 0.6 × 60 = 36

8 9 a 20 students b 37.5%

c 62.5%

10 40%

iii 21

b 2.2 × 10 = 22 d 0.21 × 400 = 84

7 a

8 a 9

b 4

c 9

d 8

11 a 49.8%

b 20%

c 1.6%

d 3.5%

e 55

f 98

g 18

h 72

e 70%

f 30%

g 15%

h 33%

i 26

j 27

k 32

l 160

9 a 8g

b $12

c $60

d 6 mL

f 14 km

g 30 L

h $18

Reasoning

e 72 kg

12 61.40% b

13 a 14 Sale 2 15 a

10 a 62% of 46

b 27% of 230

c 250% of 18

d 45% of 180

e 29% of 13

f 36% of 48

Reasoning

11 Some possibilities, student responses may vary: Problem-solving 16 55% 17 a 20

b 40%

c 40%

d 20%

18 9% 19 a 80%

b 76%

c History

20 a 30%

b 60%

c Georgia

6.04 Percentages of a quantity Understanding 1 Fraction Percentage 100% 50% 25% 20% 12.5% 10% 5% 4% 2% 1%

b 2

2 a 3 Divide by ten

818

Mathspace WA Curriculum – Year 7 mathspace.co

12 a When ﬁnding a percentage of 82 that is above 100%, the result will be larger than 82. This is because a percentage above 100% means that you are considering more than the original value (82 in this case). For example, 200% of 82 would be twice the original value, so it would be larger than 82. b When ﬁnding a percentage of 82 that is below 100%, the result will be smaller than 82. This is because a percentage below 100% means that you are considering less than the original value (82 in this case). For example, 50% of 82 would be half the original value, so it would be smaller than 82.

c i Method 1: Using fractions 60% can be expressed as a fraction:

. Now

multiply this fraction by Method 2: Using decimals 60% can be expressed as a decimal:

. Now

multiply this decimal by 82: 0.6 × 82 = 49.2 So, 60% of 82 is 49.2.

expressed as a fraction:

. Now

multiply this fraction by

g 1843.75% h 83.3%

4 a 5.2

b 2.6875

c 4.0625

d 9.6875

5 a 13.24

b 0.093

c 0.4505

d 431.9

e 33.21

f 2.8

g 49.395

h 0.24

6 a d 109.872 7 a 61

ii Method 1: Using fractions 120% can be

b 89.1268

c 9.3515

e 2.755

f 0.834 372

b 65.9

c 40

d 78

8 128 9 3 780 000 megalitres 10 66

Method 2: Using decimals 120% can be expressed as a decimal:

. Now

multiply this decimal by 82: 1.2 × 82 = 94.4

Reasoning 11 Possible answers include: • When ﬁnding 1%, 10% or 100% of a quantity.

So, 120% of 82 is 94.4

• When converting easy fractions or decimals

13 90 minutes

to percentages

14 a 28

b 28

12 a A 25% discount leaves us with 75% to pay, and a 12% discount leaves us with 88% to pay

c They give the same result. . The two expressions

d Yes. are both equal to

0.75 × 0.88 = 0.66

15 16.67% 16 38 minutes and 20 seconds 17 152 18 a 2910

b 2340

19 a 3500 people

b To verify that Betty is correct, we can multiply 0.75 by 0.88: Since the result is 0.66, Betty is indeed correct. Bernie could have just multiplied the price of the jacket by 0.66 to get the final price he needs to pay.

Problem-solving

c 5250 b 385 children

20 2625 mg 21 a $3840

e 68.75% f 124%

c Yes, Bernie would get the same result if he applied the two discounts in the reverse order. This is because the multiplication is associative, meaning that the order of the discounts does not matter. b 1.08 × 1.04 = 1.1232

13 a No b $99 840 c $1920

d 4.17%

6.05 Percentages with technology

c The manager

d Yes

e $80 870

f $80 000

Problem-solving Understanding 1 a 45 + % or 45 + Shift + % b =

c S⇔D

2 a Divide 29 by 60

b Multiply 40% by 12

c Multiply 130.8% by 9 d Divide 17 by 80%

b 67.5 kg

15 a $7.50

b $22.50

16 a 8

b 80

17 a 60%

b 40%

Chapter 6 review

Fluency 3 a 187.5%

14 a 17.5 kg

b 83%

c 60%

d 52.5%

1 a 0.10

b 0.93

c 0.234

d 0.045

Answers mathspace.co

819

e 1.6

f 4.09

2 a 3% e 620% 3 a 87.5% 4

g 7

h 78.94

b 2.1%

c 3500%

d 45.7%

f 1504%

g 0.35%

h 332.5%

b 32.14%

Fraction

c 62%

d 14.5%

Decimal

Percentage

0.07

1.51

151%

19 a 20

b 20

c They give the same result. d Yes.

× s.

20 a 2370

and 2.25

and 0.1666

and 0.625

6 a i 40%, 80%, 20%

b i 20%, 45%, 27% c i 12

%, 60%, 30% ii

22 a $4250

b $89 250 c $1716.35 d 5.88%

23 a 6.725

b 0.1025

, 0.30, 70%

> 0.40%

b 0.75 < 80%

< 57%

d 0.095 <

8 a True e True

b True

c True

7.01 Identify ratios Understanding 1 a Answers vary.

b Answers vary.

c Answers vary.

d Answers vary.

b The number of dogs being compared. c 3:5 Fluency 3 a 2 : 15 4 a i 15 : 5

ii 15 : 20

b i 7 : 13

ii 7 : 20

c i 15 : 1

ii 15 : 16

d 65%

10 a 22%

b 36%

c 60%

d 54%

11 a 35%

b 40%

c 25%

d 15%

e 68.75% f 25%

6 a 5:7

b 47 : 43

c 13 : 17

d 37 : 17

e 5 : 26

f 13 : 29

g 15 : 76

h 130 : 171

7 a 9:8 c 35%

b 5:6

ii 11 : 15

d 30%

14 35% 15 7.5% b 2

17 a 4

b 55

c 98

d 18

e 72

f 26

g 27

h 32

18 a 250% of 14

b 450% of 16

c 29% of 13

d 36% of 48

820

d 53

ii 14 : 16

9 a 12%

16 a

c 40

b i 14 : 2

h False

b 35%

h 0.24

5 a i 11 : 4

g False

13 a 20

d 2.635

g 49.395

d True

f False

12 35%

b 43

c 0.370

2 a The number of cats being compared.

d i 62.5%, 30%, 70% ii 30%, , 0.7 7 a

c 4650

21 1875 mg

24 a 43 and 0.2

b 2280

e 65.5668 f 2.8

0.625 5 a

× r.

The two expressions are both equal to

Mathspace WA Curriculum – Year 7 mathspace.co

b 100 : 31

c 200 : 409

d 21 : 120

e 360 : 659

f 43 : 52

g 37 : 24

h 13 : 120

8 a i 3:2

ii 4 : 2

b i 2 : 12

ii 4 : 12

9 a 11 : 6

b 12 : 5

c 5:8

d 3:2

e 2:1

f 50 : 1

10 a 5 : 2

b 6:2

c 5:6

d 7 : 20

Reasoning 11 The ratio of the number of oranges to the total number of fruits in the salad can be written as “oranges : total fruits”. Since the total number of fruits is 4 + 6 + 8, the ratio can be written as 6 : 18. 12 Both Lucidia and Annalee are correct. The order in which they state the length : width and the width : length are stated correctly. 13 First, we need to ﬁnd the total length of the bridge by adding all of the three lengths. So, we have the total length of the bridge is 12 m. Next, we need to ﬁnd the length of PR by adding the lengths of PQ and QR. So PR = 5 m. Finally, we can express the ratio of PR to the total length of the bridge as 5 m : 12 m. So, the ratio of PR to the total length of the bridge is 5 : 12. Problem-solving 14 3 : 8 15 a 0.08 litres 16 a 1 : 4

b 0.025 litres b 4 : 45

c 45 : 16 : 4

17 a 3 : 1 b The length will also be doubled to remain in the proportion of 3 : 1, and it will still be 3 times as large as the width. c The ratio of the area to the width is 6 : 1. This is different from part (a) where the ratio was 3 : 1. This is because the area increases with the square of the increase in dimensions, hence when the dimensions were doubled, the area became four times bigger.

7.02 Equivalent and simplified ratios Understanding 1 a 2:1=4:2 c 3 : 9 = 6 : 18 e 2 : 7 = 4 : 14 g 5 : 20 = 1 : 4

b 2 : 5 = 4 : 10 d 8 : 4 = 12 : 6 f 6 : 4 = 18 : 12 h 4 : 16 = 8 : 32

2 6 Fluency 3 $54

f 2:1 j 9:7

g 3:8 k 9 : 11

h 3:4 l 9:7

6 a Example answer: 6 : 2, 9 : 3, 30 : 10 b Example answer: 3 : 5, 9 : 15, 12 : 20 c Example answer: 4 : 10, 6 : 15, 8 : 20 d Example answer: 1 : 10, 2 : 20, 5 : 50 7 a 1:6

b 4:3

c 2 : 45

d 9:5

e 33 : 28

f 9:7

g 8 : 15

h 9:4

i 9 : 10

j 33 : 35

k 11 : 27

l 21 : 17

8 a 10 : 9

b 5:8

c 2:9

d 8 : 13

e 7:4

f 5 : 26

g 3 : 10

h 52 : 95

9 a 41 : 58

b 90 : 7

c 4 : 17

d 33 : 7

e 4:7 Reasoning 10 We can increase 1 by the same multiple to ﬁnd the number of security guards. We will divide 8400 spectators by 120 spectators to get the multiple. The multiple is 70, so multiplying 1 by 70, the number of security guards required is 70. 11 First, we will convert one of the amounts so that the units match. Let’s convert $4.00 to cents. Next, we will write the ratio and decide if we need to simplify it. 620 : 400 and the greatest common factor is 20. So we can divide both sides by 20 to simplify the ratio: 620 ÷ 20 : 400 ÷ 20. Finally, the simpliﬁed ratio is 31 : 20. This is the smallest integer valued ratio. 12 a This is true. In order for the ratio to be equivalent, increasing apples also requires us to increase oranges. b This is false. When we increase one number in a ratio, the other number must increase by the same amount. c This is true. Both numbers in a ratio increase by the same amount in order to keep the ratio equivalent. d This is false. While it is true that there cannot be any apples if there are no oranges, the opposite is not necessarily true. It is possible to have oranges without apples. 13 a The units are not the same.

4 24 5 a 2:5

e 1 : 20 i 2:3

b 1:2

c 1:2

d 5:1

c 6:7

d 12 metres

Answers mathspace.co

821

14 a 6 : 9, because it simpliﬁes to 2 : 3, while all other ratios simplify to 3 : 4 and it could be rewritten as 6 : 8 to make it equivalent to the others. b 14 : 30, because it simpliﬁes to 7 : 15, while all other ratios simplify to 1 : 2 and it could be rewritten as 14 : 28 to make it equivalent to the others. Problem-solving

So Avril needs 27 metres of green material.

15 a 16 litres

b 20 litres

c The painter will be able to paint 2 full rooms, with paint left over to paint almost half of a third room. 16 a 16

b To find the number of metres of green material, Avril can divide 33 metres of red material by the part of the simplified ratio representing red: 33 ÷ 11 = 3. Then, Avril multiply both sides of the ratio by 3 to find the number of metres of green material needed.

b 128

c No

17 Bristlecone: 600 Aleppo: 540

7.03 Find part or whole measures Understanding 1 The total parts of the ratio is the sum of the ratio parts. 2 4:3:7

11 a The student is making a mistake in interpreting the given ratio. The ratio of Kathleen’s investment to Luke’s investment is 3 : 10, which means for every 3 parts that Kathleen invests, Luke invests 10 parts. The student incorrectly assumes that Kathleen’s investment of $3000 corresponds to 3 parts and Luke’s investment of $7000 corresponds to 10 parts, but this doesn’t match the given ratio. b Kathleen and Luke invest money in a business in the ratio 3 : 7. 12 One possible way to ﬁnd the number of Gumtree trees is by writing the part to whole ratio of Gumtree trees to the total, which is 8 : 17. Then, multiply this by the total trees:

3 We can write a part to whole ratio and ﬁnd what multiple of the whole will reach the total number of blocks. Then, we can multiply each part of the ratio in order to create an equivalent ratio. The equivalent ratio will show us how many blocks Mohamad and Isabelle will receive.

Gumtree trees. Another way is by ﬁnding the number of trees in one part, that is by dividing 340 by 17 total parts, followed by multiplying 20 trees in each part by 8 parts for the Gumtree trees, which gives 160 Gumtree trees.

Fluency

Problem-solving

4 35 melons

13 a 7 parts

5 a 80 adults

b 280 children

14 27 litres

6 a $160

b $720

15 a 3 cm2

7 a 1:3

b $40

8 87 km/hr Reasoning 9 a 1:2 b 9 is 3 times of the original parts of blue paint, so multiply also 6 by 3 to get 18. 10 a 11 : 9

822

Mathspace WA Curriculum – Year 7 mathspace.co

b i 36 cm2

b 12 cm

c 9 cm

ii 21 cm2

iii 93 cm2

16 72 minutes

Chapter 7 review 1 a 24:42

b 18:54

c 90:150

2 a 3:2

b 3:1

c 7:5

3 a 3:20

b 11:12

c 3:1

4 a 1:2

b 4:3

c 4:11

5 a 12:1

b 4:1

c 5:7

6 $7000

7 21

+x +x

8 2:5

9 a 9 litres b 15 litres c The artist will be able to paint 3 full studios, with paint left over to paint a portion of a fourth studio. 10 a 36

b 132

−x −x

c No

11 Gala: 550

−x

−1

+1 +1

Reasoning

Honeycrisp: 660

10 Possible answers:

12 18 oranges

a = 1:

13 $42 14 27 km/hr 15 a 15 cm

a = 2:

b 12 cm

16 25 litres 17 25

a = 3:

18 4 cups

8.01 Introduction to variables 11 Possible answers:

Understanding 1 A variable is used to represent an unknown number.

x = 1, y = 2:

2 Possible answers: s + c + g + h or w + x + y + z 3 a q

b c and d c m and n d g

4 a 3 × a or 3a

x = 2, y = 3:

b 6 × p or 6p c 4 × m or 4 m d 3 × k + 3 × g or 3k + 3g 5 a 3a

b 8k

c 12d

d 20p

6 a p+p+p+p+p

b 5×p

c 5p

7 a 4p

b 5b

c 6y

d p + 2g

8 a x+3

b 4x + 2

c 3x − 2

d −2x + 1

x = 3, y = 4:

Fluency

9 a b

+x +x

+1 +x

12 The boxes are not the same size so the exact number of apples is not known for each box. Therefore we cannot simplify further. 13 2x + 3 means 2 lots of x and 3 lots of 1. This is exactly the same as having 3 lots of 1 and 2 lots

Answers mathspace.co

823

of x. The order when doing addition does not matter.

b 7 is added to p. c 6 is subtracted from x. d The sum of b and 9, divided by 2.

Problem-solving

9 a 5(x + 10)

14 a The total number of pens. b b = 2, r = 3 15 b = 7, g = 8, r = 9, y = 6 16 a

1 2y + 1

1 z+2

17 a b = 20

10 a 7x

b 5x + 8

c 8y − 3

d 9ab

f 4y − 9

b y subtracted from 3, then multiplied by x.

c Eleven subtracted from the product of 4 and the sum of n and 1.

4y + 2 6y + 3 10y + 5 20y + 10 3

2z + 4 3z + 6 b a = 11

5z + 10

10z + 20

c 106 oranges

8.02 Build algebraic expressions

d 3 lots of the sum of m and 7 divided by 10. 12 a i Multiplication, Division, Addition ii b i Multiplication, Addition, Subtraction ii xy + 1 c i Subtraction, Division

Understanding 1 i B

iii D

iv C

2 a 3 terms b −15

c 2

d −3

b −2

c 6

d −6

e 0.5

f 8

g 9

h −101

j −1

l π

3 a 4

ii A

ii d i Multiplication, Division, Subtraction ii e i Multiplication, Division, Addition ii f i Addition, Subtraction, Division

b −11

4 a 11

c 0 or no constant term

d 8

e –3

f –7

g –2

h 5

i 14

j –30

5 a 1

b 2

c 1

d 2

e 2

f 3

g 3

h 4

6 i D

ii A

iii F

iv B

v C

vi E

g i Subtraction, Addition, Multiplication ii 12(m + 1) h i M ultiplication, Addition, Subtraction, Division ii 13 12n

Fluency

7 a i 2

ii 8

b i 5

ii 4

c i

ii –7

d i 13

ii –13

e i –1

ii None

8 a 5 is multiplied by x.

824

d 4(m − 9) + 11

11 a The sum of b and 9, multiplied by 2.

b 3(a + b)

c 7(n − 8) + 4

Mathspace WA Curriculum – Year 7 mathspace.co

15 25 + 0.5w 16 w + 4 Reasoning 17 If we didn’t put the multiplication sign with 5 × 7 + 2 it would read 57 + 2 which is not equivalent to 5 × 7 + 2. However in algebra

3 × b and 3b are equivalent so the multiplication sign is not necessary. or 3 × k ÷ b or 3k ÷ b or

18 Any three from:

19 a Odd numbers are spaced two units apart on the number line, therefore an odd number plus any multiple of two will be another odd number. b m+6

c m + 12

e −30

f 2

g 9

h 17

i 6

j 17

k 51

l 7.2

n 22

o 13

p 5

q 38

r 58

9 a 19

b 28

c 27

d −10

e 83

f 138

10 a

d m + 2p

20 “2 groups of 8 more than x” can be written as 2 × (x + 8) while “8 more than 2x” can be written as 2x + 8.

p+7

60 40

Problem-solving 21

6p + 4

34 244

22 42 − q 23 a p × m

b L − 2K

24 a x, x + 1, x + 2

b p − 2, p − 1, p

2x + 4

5p – 3

97 11

c y − 1, y, y + 1

25 4h

11p + 5

60 126

8.03 Substitution

Reasoning

Understanding

11 a i 48

1 a A red pencil. b The apple is red. c The ﬂag is red and blue.

b I have 8 pens.

12 Any three expressions that are equal to 23 such as 4x + y

c The result of 10 − 2 is 8. d 8 is one more than seven. c 1−3

4 a x=1

b x=7

c 19

5 a 7×6+5

b 12 × 4 − 8 d

13 The smallest values of a and b is 1, therefore 10a + b = 11. The largest values of a and b is 9, therefore 10a + b = 99. Therefore 10a + b is always a two digit number. Problem-solving 14 12

Fluency 6 a 11

iv 48

d The result is also 48 because p + p = 2p and 24 × 2 = 48.

2 a Emma is 8 years old.

b 3 + 17

iii 48

c 48

d My favourite colour is red.

3 a 5×3

ii 48

b The answer is always 48 because the numbers in each expression multiply to 24 which is then multiplied by p.

15 −15 b 12

c 2

d 2

7 a 3

b 20

d −9

8 a 17

b 29

c 3

d 36

16 a i Increases

ii Decreases

b i Decreases

ii Increases

Answers mathspace.co

825

17 a i Halves

ii Doubles

b i Doubles

ii Halves

18 a i Doubles b i Multiplied by 4 19 a 5y

o 10m + 7

p 16x + 5

11 a 25m + 8n

b 8q

ii Multiplied by 3

c 10m + 10n

d 3x + 19y

ii Multiplied by 9

e 18a + 8b

f 5m + 12n

g 3ab + 2c

h 6s + 2t

b 150

20 a dp

b 3p

21 a k + m

b 20

8.04 Collect like terms Understanding 1 Like terms have the same variables raised to the same power

i 21xy + 17

j 11a + 9b − 4c

k 4x + 4y + 5

l 15m + 19n + 7

m 25x + 2y + 11

n 12y + 2

o 8x − 7y + 4z + 13

p 23j + 2k + 5

12 a 4x

b 4n

c 6m

d 8k

e 2y

f 9l

g 15h

h 21x

i 6z

j 6a

k 4b

l 8c

13 a 13x + y + 10z + 6

b 10xy + 2yz + 6xz + 3

2 A

c 16x + 2y + 41

d 11 mn + 2 m

3 B

e st + 7s + 7t + 5

f 19ab + 3bc − 9b

Reasoning

Fluency 4 a Like

b Like

c Like

d Unlike

14 1 × y is just y therefore the 1 is not needed.

e Like

f Unlike

g Like

h Like

15 5abc and −9acb are like terms, because they have the same variables raised to the same power. The order of multiplication does not matter.

5 a 10 m and 8 m, 2 and 3 b 9s and −6s c k and 4k d 10y and −y, 4 and 3 e 9a and −11a, −7b and −15b, 9 and 20 f Does not contain like terms g 11 mn and −8 mn

When x = 100, 6 × 100 + 2 × 100 − 100 = 700 and 7 × 100 = 700. Therefore 6x + 2x − x = 7x When x = 0.5, 6 × 0.5 + 2 × 0.5 − 0.5 = 3.5 and 7 × 0.5 = 3.5. Therefore 6x + 2x − x = 7x

h −17ab and −5ab, −3cd and 7cd 6 a Yes

b No

c Yes

d No

7 a Yes

b No

c Yes

d Yes

8 a No

b Yes

c No

d No

17 a i 15x

ii 20p

iii −12c

iv 1.5k

b 7 × 5 m means 7 lots of 5 m. You can multiply the numbers together to get 35 m.

ii 13u + 15

Problem-solving

b i 7u and −7u

ii 2v

18 a 2x + 6x = 8x

c i 4x and 5x

ii 9x − 6

b 3y + 3 + 5y + 5 = 8y + 8

d i 12 and −8

ii 6x + 7y + 4

c 7x − 5x + 10 − 9 = 2x + 1

9 a i 5u and 8u, 7 and 8

b 5b

d 9x + 4y + 2y = 9x + 6y

c 7 + 6a

d 9 + 7m − 2p + 3q

e 7x + 4x + 10y − 4y = 11x + 6y

e 9y + 1

f 11x − 5

f 8ab + 2b + 9b + 6ab = 14ab + 11b

10 a 7a

826

16 When x = 5, 6 × 5 + 2 × 5 − 5 = 35 and 7 × 5 = 35. Therefore 6x + 2x − x = 7x

g 14c

h 7x

i b

j 6n

k 12x

l 5xy

m 5ab

n 8b + 11

Mathspace WA Curriculum – Year 7 mathspace.co

19 a 3x + 9 + 10x − 3 + x + 6 20 12a metres 21 15y metres

b 14x + 12

22 a 8x

b 10x

23 a 15h

c 18x

9.01 Introduction to equations

b 20h

c 35h

d $70h

d 14 hours

e $560 24 a

Chapter 8 review 1 a 3 is multiplied by y c The difference of a and 5, divided by 7 b b zy + 3

3 a

1 An operator joins two equations together using an equals sign. An equation is true if the expression on the left hand side is equal to the expression on the right hand side. 2 Answer may look like:

b 4 is added to z

2 a 3p + 8

Understanding

An equation has an equals sign but an expression doesn’t.

3 Answer may look like:

To solve an equation, you need to ﬁnd the values of any variables that will make the equation true.

4 15 + 0.75p 5 2x − 6

4 Answer may look like:

6 60 − r

He uses substitution where he puts x = 6 into the equation and checks the equation is true.

7 a 19

b 9

c 35

8 a 7

b 34

c 42

9 a

Fluency 5 a Equation

b Expression

c Expression

d Equation

r+5

e Equation

f Expression

5r + 3

178

6 a 4

b 10

c 17

d 7

e 4

f 9

g 4

h 4

7 a True b False. The equation could be: 57 = 15 + 6 × 7 57 − 4 = 11 + 6 × 7

10 10

c False, the equation could be:

11 −9

5 × 7 − 19 = 16

12 4k

5 × 7 − 22 = 13

13 a 3 + 8b

b 10xz

14 a 16x + 2

b 4p + 11q − 6

15 a 52x

b 13.5a

16 a ru + 9r + 9u + 5

c 10cd

c 15c b 19cd + 6de − 10d

17 16x metres

d True 8 a x = 12

b x = 12

c x=7

d x = 27

e m = 13

f f = 23

g u=6

h m = 48

9 Answers may vary a The product of eight and x is equal to thirty two.

18 15y

b The quotient of sixty three and x is equal to nine.

19 $95k

c The product of ﬁve and y is equal to sixty.

20 a

b 30 hours

d The quotient of seventy two and p is equal to eight. e The product of six and x plus four is equal to sixty minus twenty.

Answers mathspace.co

827

f T he product of eight and x plus seven is equal to seventy two minus twenty ﬁve.

9.02 Write rules for relationships Understanding

b m – 16 = 26

10 a c q – 12 = 25

d 7y = 25 – x

f 5q = 29 – p

g 65 – 7p = 12

h 72 – 12r = 36

11 a Yes

b No

c Yes

d No

e Yes

f Yes

g No

h Yes

12 a No

b Yes

c No

d No

e No

f No

g Yes

13 a No

b No

c Yes

e No

f Yes b No

c No

d No

e No

f No

g Yes

h Yes

m−17

b m = 29 16 a

b Decreasing by 2

c Increasing by 7

d Decreasing by 5

2 a The coeﬃcient is 3

b The coeﬃcient is −2

c The coeﬃcient is 3

d The coeﬃcient is −4

Fluency 3 a

Figure number (F )

Number of matchsticks (M)

c M = 3F

b 3 d No

14 a Yes

15 a

1 a Increasing by 1

4×k

96 100

b k = 24 Reasoning 17 a Left-hand side b Try a number larger than 10. c Right-hand side d The solution to the equation lies between 10 and 20. 18 a Right-hand side b Try a number larger than 30. c Left-hand side d The solution to the equation lies between 30 and 40. Problem-solving

4 a

Figure number (F )

No. matchsticks (M)

b Increasing by 1

c M=F+4

5 a 3 b The number of matchsticks increases by three c The number of matchsticks is equal to 3 times the number of squares plus 1. d M = 3F + 1 6 a N = 2F + 1 b 25 tiles 7 a T = 3W−2 b T = 298 tiles 8 a i y increases by 1

ii y = x + 8

b i y increases by 1

ii y = x − 5

c i y increases by 7

ii y = 7x

d i y increases by 5

ii y = 5x + 2

e i y decreases by 3 ii y = −3x + 17 f i y increases by 2 9 a y=x−3

ii y = 2x − 5 b y = 5x

c y=x+7

d y = 2x + 2

e y = 3x − 13

19 C

Reasoning

20 C = 20 + 12h

10 Method 1: continue the pattern of adding 4 to the b values and the missing number is 14 + 4 = 18.

21 10p = 30 22 b + 16 = 42

828

Mathspace WA Curriculum – Year 7 mathspace.co

Method 2: Find the rule for the table b = 4a − 10 and use it when a = 7 to ﬁnd b = 4 × 7 − 10 = 18.

11 Yes the rules are the same as it doesn’t matter what order you do addition in. 12 a Each triangle has 3 sides therefore we multiply by three. b M = 4S, where s is the number of squares. c M = 6H, where H is the number of hexagons. Problem-solving 13 a

No. of bunches (x)

No. of bananas (y)

b The number of bananas is equal to 4 times the number of bunches. c y = 4x 14 a

No. of decks (d)

No. of cards (c)

104

156

208

b The number of cards is equal to 52 times the number of decks. c c = 52d 15 a The number of marbles will be equal to 4 times the number of bags plus his original 5. b M = 4b + 5 16 a

Week (W)

Account $200 $215 $230 $245 $260 total (A) b A = 15W + 200 17 a

g Divide by 4

h Divide by 7.3

i Subtract

j Add

k Subtract

l Add

4 a x = 50

b y=9

c b = 35

d k=7

5 a x = 50

b y=9

c b = 62

d k = −18

6 a x = 30

b d = 12

c x=9

d a=7

7 a x = 24

b y = 36

c m = 200 d x = 0

8 a q=2

b s = 21

c r = 47

d u = −1

e v = 48

f u = 26

g k=5

h k=7

i r = 20

j q = 42

k y = −11

l p = 35

mx=0

n n=5

o x = 78

p m = −5

9 a x = 1.8

b x = 6.6

e x=0

f b = 12

10 a i 10 + p = 12

ii p = 2

b i −9 + p = 21

ii p = 30

c i p − 20 = 5

ii p = 25

d i p − 13 = −15

ii p = −2

e i p + 39 = 83

ii p = 44

f i 75 = p + 42

ii p = 33

g i p − 17 = 25

ii p = 42

h i

ii p = 96

i i 10p = 60

ii p = 6

j i

ii p = 72

Figure number

k i 3p = 18

ii p = 6

No. of tiles used

l i 12p = 84

ii p = 7

ii p = 121

n i 7p = 84

ii p = 12

b The number of tiles used are the square numbers. c N=F2

d 49 tiles

Reasoning

9.03 One-step equations Understanding 1 a Addition

b Multiplication

2 a Divide by 8

b Subtract 8

c Multiply by 8

d Add 8

11 Answer may look like: I would subtract 5 from both sides of the equation. This is because the goal is to isolate the variable (x) on one side of the equation, and subtracting 5 from both sides will eliminate the constant term 5 from the left side. 12 Answer may look like:

Fluency 3 a Divide by 7

b Add 2

c Subtract 3

d Add 4.5

e Divide by 4.1

f Add 1.39

Yes, it is possible to solve this equation by adding 8 to both sides. Since the constant term on the right side is negative, adding 8 to both sides will eliminate it from the left side.

Answers mathspace.co

829

13 a 30h = 300

b 10 hours

c 4 hours

d Answer may look like: There could be several reasons why Samantha might charge more or less per hour for her freelance work compared to Lisa. It could be based on factors such as their experience, skills, expertise, location, demand for their services, etc. Samantha may have more experience or expertise in a particular area, or she may be in higher demand than Lisa. Alternatively, Lisa may live in a location where the cost of living is lower, allowing her to charge less per hour.

5 a

b c

15 a The company needs to hire 50 new under 35 year-old employees.

c m − 25 = 44, m = 69 d h + 56 = 80, h = 24

b The percentage of the total workforce that will be under 35 years old after the new hires will be 45.45%. d

Problem-solving 14 a 10n = 80, n = 8

9.04 Two-step equations 28

Understanding 1 a Take shoes off then take socks off.

b ÷6 then + 8

c +2 then × 5

d ×9 then − 5

3 a −3

b +7

c ÷2

d ×5

3 (6 3) 2

2 9

5y 10

Solve using the inverse operations.

4 6

830

5y 10

4 a

ii Determine the inverse operations.

Fluency

3 2 4

6 a i

d Finish eating lunch then put lunch box back in bag. 2 a −11 then ÷ 3

b Turn lamp on then get out of bed. c Stop watching movie then turn off TV.

Mathspace WA Curriculum – Year 7 mathspace.co

b i

q 3

5 5(q 3)

ii Determine the inverse operations.

5(q 3)

q 3

e i

Solve using the inverse operations. ii Determine the inverse operations.

c i

s 9

Solve using the inverse operations.

6(s 9)

ii Determine the inverse operations.

6 s 9

6(s 9)

Solve using the inverse operations.

f i

9 8y 9

ii Determine the inverse operations.

y 8

d i

ii Determine the inverse operations.

8y 9

Solve using the inverse operations.

n 8

7 a 2n + 6

b 2n + 6 = 12

Solve using the inverse operations.

Answers mathspace.co

831

c Subtract 6 from both sides then divide both sides by 2.

Problem-solving 19 44 kilometres

20 12 hours

c Multiply both sides by 5 then subtract 8 from both sides.

21 38 months

8 a

23 8 months

9 a

c Subtract 5 from both sides then multiply both sides by 2. 10 a w = 14

22 40 weeks

b x=4

24 a A = 27 m2

b width = 5 m

25 c = 10 cm 26 a F = 95

b C = 100

Chapter 9 review

c m=3

d y=5

1 a No

b Yes

c No

2 a Yes

b No

c Yes

3 C 4 b + 12 = 39

5 a y = 3x + 2

b y = 8x − 3

6 a Divide by 3

b Subtract 8.5 c Add

e x=2

f t=6

g m=1

h p = 10

i x = 40

j x = 15

k p = 66

l n = 72

11 a u = 3

b q = 34

c m=9

d t = 37

7 a x = 30

12 a t = 3

b p = 15

c x=1

d x=4

d 17

f s = 23

g p=3

h s = 12

8 a x = 43

e n = 22 13 a t = 6

b z=5

c x = –12

d p=0

e x = 42

f n = 32

g t = 45

h s=3

i s=2

j x=5

k x=4

l x = 21

14 a

d Divide by 4

d 14 9 a

e Add 5

f Subtract 7

b x = 20

e 16

f 56

b x = −24

c x = 9.4

e 13

f 18

c x=0

10 a i 15n = 225

ii n = 15

b i n − 19 = 76

ii n = 95

c i

ii n = 288

11 a 6c = 90, c = 15 b 18

c b − 20 = 30, b = 50

Reasoning

12 a a = 3

b b=4

c c=3

15 Answers may vary

13 a x = 4

b m=3

c n=3

16 Answers may vary

14 a x = 50

b y = 27

c z = 25

17 Subtract B from both sides then divide both sides by A

15 a a = 0

b b=3

c c=0

16 a x = 4

b y=9

c z = 13

18 a Divide both sides by C then add D to both sides b

832

Mathspace WA Curriculum – Year 7 mathspace.co

17 a The number of circles in the next step is obtained by adding 4 circles to the previous step.

b 17

8 a

c C = 4S − 3

7 6

19 35 kilometres

21 a

4 3

Figure number

No. of Circles

c C = 4F − 1

b Increasing by 4 22 w = 13.2 cm

b F = 59 2

24 a A = 12 cm

b y = 11 m

25 a P = 22 cm

b w = 5 cm

x 1

2 3 4 5 6 7 8 9

b Point C

c Point B

b y-coordinate

c 9

10 a A (5, 2), B (5, 8), C (7, 2) b A (5, 3), B (4, 7), C (7, 7), D (8, 3)

10.01 The Cartesian plane Understanding

11 a (6, 0)

b (0, 4)

e (2, 2)

f (5, 7)

c (13, 6)

d (0, 8)

12 B

1 (2, 5), (2, 0)

13 8 units

2 (0, 3), (6, 3)

14 2 units b False

c True

d True

Fluency 4 a Star

b Pig

c Dog

d Bird

5 a (7, 7)

b (9, 4)

c (1, 0)

d (10, 10)

6 a (2, 0)

b (0, 5)

c (7, 1)

d (3, 4)

e (3, 3)

f (5, 8)

L M

8 B 7 6 I

E K

15 a (8, 6)

b (0, 6)

c (4, 10)

d (4, 2)

16 a 6 units

b 7 units

c 4 units

d 8 units

17 a

10 9 8 7 6 5 4 3 2 1

y A

x 1 2 3 4 5 6 7 8 9 10

9 a A (0, 9), B (1, 8), C (2, 7), D (3, 6), E (4, 5)

23 a C = 20

3 a True

20 q = 12 m

18 6

b 8 units

c 4 units

3 2 P 1 F

D H N A 1

2 3 4 5 6 7 8 9

Answers mathspace.co

833

18 a

y 10 9 8 7 6 5 4 3 2 B 1

10.02 Directed numbers on the Cartesian plane

Understanding

1 2 3 4 5 6 7 8 9 10

19 a

10 9 8 7 6 5 4 3 2 1

ii 0

b i 0

ii 0

2 y-axis

b 7 units

1 a i 0

3 b=0 4 a 3rd quadrant, 4th quadrant

c 6 units

b 2nd quadrant, 3rd quadrant

c 1st quadrant, 3rd quadrant P

d 2nd quadrant

e 3rd quadrant R

5 a (0, −9)

b (−7, 6)

e (−2, 1)

f (−3.5, 0)

d (4, 3)

6 B x 1 2 3 4 5 6 7 8 9 10

b 3 units

c 7 units

d Trapezium

7 1 8 a

10 8 6 4 2

Reasoning 20 a 7 units to the right b 10 units upwards d 6 units to the left and 4 units down 21 a i Move 5 units to the right then 5 units down ii Move 8 units to the left then 9 units up b Kite 22 a 4 metres

b 9 metres

c 21 metres

23 Go c − a units to the right and b − d units down Problem-solving 24 a A (5, 15) b B (1, 7)

c C (4, 16) d 4 m

25 a 200

b 700

26 a i 2 metres

c 9 hours d 1000 ii 6 metres

b 6 metres c 9 metres

Mathspace WA Curriculum – Year 7 mathspace.co

x 2 4 6 8 10

b (−4, 4) Fluency 9 a (6, 7)

b (3, 9)

c (4, 4)

d (9, 5)

e (−4, 3)

f (−9, 6)

g (−2, 7)

h (−7, 8)

i (−2, −8) j (−6, −7) k (−7, −2)

l (−5, −9)

m (3, −2) 10 a Q

e 16 m

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

c 5 units to the right and 4 units down

834

c (6, −2)

n (4, −7)

o (8, −6)

p (7, −1)

b S

c P

d R

11 a (1.5, 3) c (−2.5, −3.5) 12 a A (3, 0)

b (−1.5, 0) d (3.5, −3.5) b B (−2.5, 0)

c x-axis

9 y 8 N 7 6 G 5 P 4 3 O B I 2 T 1 S A F x −9 −8 −7 −6 −5 −4 −3 −2 −1−1 1 2 3 4 5 6 7 8 9 −2 −3 M R D C E Q −4 −5 J −6 −7 H −8 L −9 K

14 a 1st quadrant

b 2nd quadrant

c 4th quadrant

d 3rd quadrant

15 a i

5 4 3 2 1

d i

−1−1 −2 −3 −4 −5

x 1 2 3 4 5 6 7 8 9

ii 4th quadrant e i

4 2 1 −9 −8 −7 −6 −5 −4 −3 −2 −1 −1

−3 −4

ii 3rd quadrant f i

3 1

x 1

−1

−2 −3 −4

g i

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

ii 4th quadrant

ii 2nd quadrant 5 4 3 2 1

−4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

−4 −3 −2 −1

x 1

−2

1 2 3 4 5

ii 1st quadrant b i

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

c i

5 4 3 2 1

2 1

−4 −3 −2 −1 −1

−2

1 2 3 4 5

−3 −4

ii No quadrant

ii 3rd quadrant

Answers mathspace.co

835

h i

5 4 3 2 1

d i

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

17 a A (5, 9), B (5, −3), C (−2, −3), D (−2, 9)

b T (8, 6), U (8, −9), V (−4, −9) c A (6, 2), B (−9, 3), C (−9, −7), D (6, −8) x

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

2 4 6 8 10

d S (8, 9), T (−5, 4), U (−7, −3), V (6, −6) 18 a

8 4 2

−8 −6 −4 −2 −2 10 8 6 4 2

−8 x 2 4 6 8 10

3 units b

6 B

ii Rectangle

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

−6

−8 −6 −4 −2 −2 10 8 6 4 2

−4

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

6B 8

−4

−6 −8 x 2 4 6 8 10

ii Parallelogram

8 units c

8 4 2 −8 −6 −4 −2 −2 −4 −8

4 units

Mathspace WA Curriculum – Year 7 mathspace.co

−6

836

ii Square

c i

2 4 6 8 10

ii Rhombus 10 8 6 4 2

b i

−10−8 −6 −4 −2 −2 −4 −6 −8 −10

1 2 3 4 5

ii No quadrant 16 a i

10 8 6 4 2

27 a

y 10 8 6 4 A 2

6 4 A

−8 −6 −4 −2 −2

−10−8 −6 −4 −2 −2 B −4 −6 −8 −10

−4 −6

−8

b 5 units

9 units 19 a

d 20 units2

c 8 units

Understanding

1 a (−3, 7), (−2, 5), (−1, 3), (0, 1)

x 2

b (−2, −2), (−1, 0), (0, 2), (2, 4)

2 a Linear

−4 −6

b Non-linear

c Non-linear

3 Four points that form a linear relationship are A, R, O, B.

−8

b 3 units

10.03 Graphs of linear relationships

6 2 C −8 −6 −4 −2 −2

2 4 6 8 10

c 7 units

Fluency 4 a i

Reasoning

20 a Move 2 units to the right, and then 4 units down. b Move 9 units to the right, and then 14 units up. 21 a Move 8 units to the left, and then 3 units up. b Move 4 units to the right, and then 10 units up. c Move 6 units to the right, and then 5 units down. d Move 6 units to the left, and then 11 units up.

6 5 4 3 2 1

x 1

ii Yes

Problem-solving 22 a 12 m

b 8m

c 14 m

d 20 m

b i y 3

23 (−2, 2)

24 (−2, −1)

25 a Any point on the line y = −1, except x-coordinate of −3.

−4 −3 −2 −1 −1 −2

b Points should lie on y = 2 or y = −4, except x-coordinate of −3. 26 a (−3, 1)

b (−5, 1)

x 1 2 3 4 5 6 7 8 9 10

−3 −4

c (−3, −5) d (9, 1) ii No

Answers mathspace.co

837

c i y

8 7 6 5 4 3 2 1

2 1 −2 −1

x 1

−1

−2 −3

−3 −2

−4

−1 −1

−2 −3 −4

−5

ii Yes

b i

d i

y 6 5

−1

1 −6 −5 −4 −3 −2

x 1

−1

−2

ii Yes

−3

e i y 6

c i

5 4

x 1

8 6

ii No f i 8 y 7 6 5 4 3 2 1

−5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8

5 a i

1 −1

−4 −3 −2

Yes

x 2

d i

1 2 3 4 5

−2

−1

y 10 8

−1

−4

4 2 x −2

838

Mathspace WA Curriculum – Year 7 mathspace.co

−1

6 a

10 a

14 12 10

2 1

b Yes

6 4 2

−2

11 a

−1

−3 −4

c 2 y

9 8 7 6 5 4 3 2 1

x 1

Number of matchsticks ( y) 5

9 8 7 6 5 4 3 2 1

x 1

−4 −3 −2 −1−1

Figure number (x)

−2

b Yes

c Linear

1 −4

t 1

Reasoning

−2 −3

12 2 points

b Yes

13 Any three points with the same x and y-values.

c 13

Weight (kg)

Total Cost ($)

Height of pot plant (y cm) y

c 20

7 a

9 a

−1

8 a

14 12

−6

−8

−2

Time (t weeks)

14 No, because the two points with x-coordinate of −3 are in a vertical line, and the other point with x = 1 is not on the same vertical line. Therefore they do not lie in a straight line.

15 Yes

Problem-solving

16 a

Week

No. of apples

5 x 0

c Linear

Answers mathspace.co

839

Apples

8 7

No. of steps

Length of staircase (cm)

6 5

Length 30

4 3

2 1 0

Week 5

d y = 2x

c Yes 17 a y = −2

Steps 0

b y=4

c y = −17

d x=5

Figure number

No. of matchsticks

21 a

b 13 Matchsticks

Step

d y = 3x

c Yes

−2

Surface air temperature −4 in °C, y

−3

20 a

−1

Revenue, r

22 20 18 16 14 12 10 8 6 4 2

d Yes. The relationship is linear. e $8

−2

−1

d y = x−1

c 13 °C No. of steps

Height (cm) 45

22 a

Height of staircase (cm) b

Number of loaves, l

1 2 3 4 5 6 7 8 9 10

Water temperature −3 in °C, x

b 3 °C

i Yes

b B

12 11 10 9 8 7 6 5 4 3 2 1

Time (t mins)

Capacity of water (C cm)

10 9 8 7 6 5 4 3 2 1

c Yes

t 1

Number of steps 2 3

d y = 15x

e Yes

Mathspace WA Curriculum – Year 7 mathspace.co

d Linear

c B

840

h Yes

18 a

19 a

23 a M = S+7 b i Yes

ii No

iii Yes

v Yes

vi Yes

vii No

iv No

b i

25 20 15

5 S 5

1 x-axis

11 a 3 units

2 a=0

12 a

3 a 1st quadrant

b 4th quadrant

c 3rd quadrant

d 2nd quadrant

b 8 units 9 8 7 6 5 4 3 2 1

4 B 5 a (8, 0)

b (0, 5)

e (2, 5)

f (7, 9)

6 a (11, 8)

b (1, 8)

10 9 8 7 6 5 4 3 2 1

2 4 6 8 10

ii Rhombus

Chapter 10 review

7 a

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

10 8 6 4 2

c (17, 7)

d (6, 10)

x 1

b Yes 13 a A

−2 −3

d (6, 3)

d 9 units

−4 −3 −2 −1−1

c (6, 13)

c 4 units

c 10

Time (t minutes)

Temperature (y °C)

16 14

1 2 3 4 5 6 7 8 9 10

b 3.6 units

c 3.6 units

d Parallelogram

8 a (2, 4)

b (−2, 1)

c (−3, −4) d (4, −4)

9 a Q

b S

c P

10 a i

10 8 6 4 2 −10 −8 −6 −4 −2 −2 −4 −6 −8 −10

4 2

d R

t 1

c Linear 14 a i 1 metre

ii 2.5 metres

b 5 metres x 2 4 6 8 10

c 5.5 metre 15 (−3, 3)

ii Square

Answers mathspace.co

841

11.01 Geometry notation and language

16 a 19 Pages 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Understanding 1 a Line

b Ray

c Segment

d Line

e Ray

f Segment

g Line

h Ray

2 a

Hour 1

G F

b Yes c

c y = 4x + 1

17 a

Hours (h)

Widgets produced (n)

b B c

10 9 8 7 6 5 4 3 2 1

f E

B A

h 2

3 a The line through A and B. b The segment between A and B. c The ray from B through A. 4 a False

18 a O = E + 6 b i Yes

ii No

iii Yes

iv No

v Yes

vi Yes

vii No O

b True

c False

d False

5 a PR and PQ b AB and AE BC and DC Fluency

6 a • The line through Q and S.

• The line through X and S.

b • The ray from C through P.

• The ray from C through X.

7 The ray from P through Q.

5 E 0

d Non-linear

842

d Q

Mathspace WA Curriculum – Year 7 mathspace.co

8 ∠TMG 9 a ∠ADC

b ∠BCD

c ∠ABC

e ∠XWZ

f ∠WZY

g ∠EAB

d ∠WXY

10 a ∠BCD and ∠DAE

b ∠BCD and ∠BAD

c ∠BAD and ∠BCD

d ∠ABC and ∠BCD

e ∠BAD and ∠ADC

f ∠ABC and ∠ADC

4 a Vertically opposite angles

Reasoning 11 If point Q lies on line PR, it means that Q is a point on the line PR. This shows that Q, R, and P lie on the same straight line, though not necessarily in that order. 12 The naming of rays differs from the naming of line segments because rays have a direction, starting from their endpoint and extending indefinitely in one direction. When naming rays, the first letter represents the endpoint, and the second letter represents another point along the ray to indicate its direction. a A laser pointer is a model for a ray. Light is emitted from the laster pointer, and then travels in a straight line off indefinitely, only stopping if it hits an object. b The Golden Gate Bridge is a model for a line. Traffic can drive across it in both directions. It doesn’t extend infinitely like a true straight line, however. Also, it curves very slightly due to the curvature of the Earth.

14 5

5 a Close to 60°; acute b Close to 120°; obtuse c Equal to 90°; right d Close to 120°; obtuse 6 a Complementary angles

16 a 8

b 18 B

11.02 Measure angles Understanding 1 a No

b Yes

c No

d No

2 a True

b False

c True

d False

e True

f False

Neither

c Neither d Supplementary angles b ∠BXC and ∠EXF c ∠AXF and ∠EXF or ∠AXF and ∠BXC 8 a ∠AXB b ∠AXD or ∠BXC Reasoning 9 a These angles are not complementary angles because 68° + 12° = 80° and complementary angles must add up to 90°.

10 No, we cannot say that the angles are vertically opposite because there is not enough information. The angles share a common central point and are the same size, but vertically opposite angles are only formed from intersecting lines. It is possible for two angles to share a central point and have the same measure but not be vertically opposite, such as the angles in this diagram: A C

Fluency 3 a Obtuse angle

b Right angle

c Obtuse angle

d Acute angle

e Straight angle

f Acute angle

g Reflex angle

h Right angle

i Acute angle

j Obtuse angle

k Revolution

c These angles are supplementary since 109° + 71° = 180°.

15 10

d Complementary and adjacent angles

b These are vertically opposite angles because they are opposite angles formed by intersecting lines and have equal measures.

Problem-solving

c Supplementary and adjacent angles

7 a ∠BXC and ∠CXD or ∠AXF and ∠EXF

13 Example answers:

b Adjacent angles

Answers mathspace.co

843

11 There are three angles with a central point at Q, so ∠Q could refer to any one of the three angles shown in the diagram. The question should have named the angle with three points, so the students would not be confused about which angle they needed to find.

7 Corresponding angles 8 a i ∠EPQ

ii ∠APQ

iii ∠APF

b i ∠CQP

ii ∠EQP

iii ∠DQE

9 a x = 110° since these are corresponding angles b x = 75° since these are alternate angles c x = 55° since these are co-interior angles

Problem-solving b x = 40°

10 a Angle 5

c x = 97°

d x = 86°

c Angle 4

d Angles 1, 4, 5, 7

e x = 130°

f x = 75°

e 105°

f 85°

g x = 25°

h x = 20°

g 140°

13 a x = 60°

b y = 72°

11 a Angle 4 b Angle 3 c 124°

c z = 10°

d x = 75°

12 a 5, 9 and 14

e x = 40°

f x = 40°

12 a x = 50°

11.03 Parallel lines and transversals

b Angle 4

d 113°

b 8, 12 and 15

c 9 and 14, 10 and 13, 11 and 16, 12 and 15 d 112°

e 105°

f 75°

13 A; vertically opposite to the corresponding angle of ∠NBP

Understanding 1 a Parallel lines are indicated using arrows or ‘chevrons’. b Example answer:

14 C; alternate angles Reasoning 15 a The diagram contains parallel lines. The marked angles are supplementary and co-interior so we have a transversal of parallel lines.

2 a ‘C’ shape

b ‘Z’ shape

3 a Equal

b Supplementary

c The diagram contains parallel lines. The marked angles are equal and corresponding so we have a transversal of parallel lines.

c Equal Fluency 4 a Non-parallel c Non-parallel 5 a Alternate angles

b Parallel d Non-parallel b Co-interior angles

6 a x = 116: The angles are co-interior angles, so they are supplementary. This means they add to 180°, so, 64 + x = 180 64 + x − 64 = 180 − 64 x = 116 b x = 107: The angles are corresponding, so their measures are the same. c x = 96: The angles are alternate angles, so their measures are the same.

844

b The diagram does not contain parallel lines. The marked angles are not supplementary so they are not co-interior. We therefore do not have a transversal of parallel lines.

Mathspace WA Curriculum – Year 7 mathspace.co

d The diagram does not contain parallel lines. The marked angles are not equal so they cannot be corresponding. We therefore do not have a transversal of parallel lines. e The diagram contains parallel lines. The marked angles are equal and alternating so we have a transversal of parallel lines. f The diagram contains parallel lines. The marked angles are not equal so they cannot be alternating. We therefore do not have a transversal of parallel lines. g The diagram does not contain parallel lines. The marked angles are not equal so they cannot be corresponding. We therefore do not have a transversal of parallel lines.

h The diagram contains parallel lines. The marked angles are supplementary and co-interior so we have a transversal of parallel lines. 16 It is not possible to say if the lines are parallel as there is not a matching pair of alternate or corresponding angles, or a pair of supplementary co-interior angles. 17 a p = 64° b Example answer: Angle p and the angle marked 26° are adjacent angles. Together, they form a right angle. This means p + 26 = 90 p + 26 − 26 = 90 − 26 p = 64° c q = 64° d Example answer: The line at the bottom of the diagram is a transversal that passes through the two parallel lines shown. This means angles p and q are corresponding angles, so their measures are the same.

25 a x = 29°

b y=

26 x = 80°, y = 47°, z = 100°

11.04 Triangles Understanding 1 180° 2 a Its interior angles are all acute or less than 90°. b One of its interior angles is obtuse or more than 90°. c One of its interior angles is a right angle or exactly 90°. 3 a All of its interior angles have different sizes and all its sides have different lengths. b At least two of its interior angles are the same size and at least two of its sides are the same length. c All of its interior angles are the same size and all its sides are the same length. This implies all its interior angles are 60°. Note that an equilateral triangle is also isosceles. 4 a ∠ABF = ∠BCD + ∠CDB

Problem-solving 18 a They are alternate angles, so they are also equal. b They form a straight angle, so they are also supplementary. c 96° 19 a They are vertically opposite angles, so they are also equal. b They are co-interior angles, so they are also supplementary.

∠BDE = ∠DBC + ∠DCB ∠CDF = ∠DBC + ∠DCB b Exterior angles CDF and BDE are both exterior angles of angle BDC. They are also vertically opposite angles, which means they are equal. Fluency 5 a Right-angled, isosceles triangle b Acute, isosceles triangle c Acute, equilateral triangle

c 73° 20 a They are co-interior angles, so they are also supplementary.

d Obtuse, scalene triangle e Acute, isosceles triangle

b They form a straight angle, so they are also supplementary.

f Right-angled, scalene triangle

c 97°

h Obtuse, isosceles triangle

21 x = 39 22 a x = 114°, y = 54

b x = 40, y = 50

23 a x = 117°

b y = 117°

24 a x = 33°

b y = 57°

g Acute, scalene triangle 6 a Isosceles

b Scalene

c Equilateral

d Scalene

e Equilateral

f Isosceles

g Isosceles

h Isosceles

c 180°

Answers mathspace.co

845

7 a Acute

b Obtuse

c Acute

d Right-angled

e Obtuse

f Obtuse

8 a 50°

b 90°

c 27°

d 50°

9 a i x = 70 b i x = 110 ii Obtuse, scalene triangle b x = 40

e x = 126

f x = 68

c x = 69

d x = 56

Reasoning 11 A different way to find x is by using the exterior angle to find the measure of the third interior angle. Since the third interior and the exterior angle form a straight angle, they are supplementary:

b True c False, as seen in any acute triangle. The sum of the measures of the interior angles of a triangle is 180°.

ii Acute, scalene triangle

10 a x = 55

15 a False, as seen by finding the sum of any non-right triangle, such as an equiangular triangle. The interior angles of any triangle sum to 180°.

180 − 108 = 72

d False, as seen in any right triangle that is not an isosceles right triangle. In a right triangle, the right angle is 90° and the remaining two angles are always complementary. They are only equal in an isosceles right triangle. Problem-solving 16 a x = 67

b x = 62

c x = 49

17 a x = 74, y = 43

b a = 61, b = 29

c x = 61, m = 138

d a = 60, b = 13

18 a x = 64, y = 32

b x = 38, y = 32

c x = 59, y = 121

d x = 60, y = 65

Then, we can set the sum of the three interior angles equal to 180: x + 52 + 72 = 180 x + 124 = 180 x + 124 − 124 = 180 − 124 x = 56

20 a They are corresponding angles.

12 The sum of the angle measures exceeds 180°.

21 a They are alternate interior angles.

13 Yes, this is true. In the diagram, the interior angles ∠1 and ∠2 have a sum that is equal to two different exterior angles, ∠3 and ∠4. The exterior angles are vertically opposite angles, so they are also equal. 1

2 3

d x = 44

19 Yes, Jena has pitched her tent correctly. b They are vertically opposite angles. c x = 41 b x = 24

11.05 Quadrilaterals Understanding 1 a False

b True

c False

d False

e True

f True

g True

h False

2 Yes, a trapezium. 3 CD 4 a Concave b Convex c Concave d Convex

14 The smallest exterior angle has a measure of 95°. The third internal angle has a measure of 180 − 37 − 85 = 58 and we know the measure of an exterior angle of a triangle is equal to the sum of the interior angles on the opposite side. The smallest exterior angle will therefore be the sum of the two smallest interior angles which gives us 58 + 37 = 95°.

e Convex Fluency 5 a x = 30

b x = 110

c x = 230 d x = 106

e x = 66

f x = 65

g x = 63

6 a Kite b Rectangle, parallelogram c Rhombus, kite, parallelogram d Kite

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Mathspace WA Curriculum – Year 7 mathspace.co

h x = 87

e Rectangle, square, rhombus, kite, parallelogram

14 4, this is the maximum number of angles a quadrilateral can have, and it would give the correct angle sum. Squares and rectangles have four right-angles.

f Kite g Rectangle, parallelogram h Parallelogram i Trapezium 7 a Trapezium

b Neither

c Parallelogram

d Trapezium

8 a Any parallelogram that doesn’t have any right angles, and consecutive sides are different lengths. For example: B

15 100°

12 80°

b Any figure that is a rhombus. For example: 30°

30°

c Any figure that is a rectangle (including squares). For example: D

10 A

16 a The quadrilateral is a square. A quadrilateral with one right angle and four congruent sides is a square by definition. b The quadrilateral is a rhombus. A quadrilateral with four congruent sides is a rhombus by definition. c The quadrilateral is a parallelogram. A quadrilateral with opposite sides and angles congruent is a parallelogram. d The quadrilateral is a rhombus. A quadrilateral with opposite sides parallel and consecutive sides congruent is a rhombus.

8 cm 8 cm

15 All parallelograms have opposite sides that are equal in length. Since we are not told anything about the angles or the exact lengths of all four sides, parallelogram is the most precise classification.

17 It is because some quadrilaterals share the same properties.

Problem-solving

18 a x = 19

19 ∠CDE = 24°, use the fact that ∠CDA = 90° so ∠EDA = 90° - 24° = 66°.

b x = 104

c x = 135

d x=5

9 AB and CD

Then, use the fact that the adjacent angles of a rectangle are equal so:

10 a ∠a = 77° ∠b = 103° ∠c = 103°

b ∠a = 69° ∠b = 69° ∠c = 111°

66°

c ∠a = 120° ∠b = 120° ∠c = 60° d ∠a = 105° ∠b = 75° ∠c = 75° 11 a x = 99, y = 107 b a = 63, b = 63, c = 117 c a = 79, b = 101 d p = 5, q = 9 Reasoning 12 1, because otherwise the angle sum would be larger than 360. 13 3, because otherwise the angle sum would be larger than 360.

B 24° 66°

24° E

66° 24°

24°

66° C

20 x = 2 m

Chapter 11 review 1 a

b X

P Y

Answers mathspace.co

847

2 a • The line through B and D.

12.01 Units of length

• The line through B and X. Understanding

b • The ray from Q through B. • The ray from Q through O. 3 a ∠MNO and ∠OLQ

b ∠BAD and ∠BCD

c ∠ABC and ∠BCD 4 a Parallel

b Non-parallel

c Non-parallel

1 a m

b km

e cm

f m b 100 cm = 1 m

c 1 cm = 10 mm

d 1000 mm = 1 m

3 a Divide

5 a False

b False

4 a 100

b 10

c True

d False

e 100

f 10

b Multiply c 1000

d 1000

Fluency

6 a i ∠BQD

ii ∠FQB

iii ∠FQE

b i ∠DPB

ii ∠DPF

iii ∠FPA

b x = 74°

c x = 78°

7 a x = 125°

d mm

2 a 1000 m = 1 km d Parallel

e False

c cm

8 x = 38° 9 a x = 34°

b y = 56°

10 a x = 21°

b y = 17°

5 a 30 mm

b 6 cm

c 600 cm

d 8m

e 6000 m

f 9 km

g 8.82 cm

h 733 cm

i 0.149 m

j 12.226 km

k 46 mm

l 5690 m

6 a 24 000 mm

b 3.484 m

11 a Right-angled, isosceles triangle

c 600 000 cm

d 8.668 km

b Obtuse, isosceles triangle

e 9 000 000 mm

f 1.5284 km

c Acute, equilateral triangle

g 9030 mm

h 0.15 m

d Obtuse, scalene triangle

i 743 000 cm

j 0.453 km

e Acute, isosceles triangle

k 6 710 000 mm

l 0.32 km

f Right-angled, scalene triangle g Acute, scalene triangle h Obtuse, isosceles triangle

7 5.068 km 8 12 500 cm

12 a x = 30

b x = 37

9 Nadia is shorter.

c x = 70

d x = 128

10 a Haonan

e x = 70 b x = 210

c x = 77

d x = 56

12 Plant 1 13 a

Name

14 a x = 81, y = 73 b a = 61, b = 61, c = 119

One World Trade Center

15 x = 24° 16 a x = 73, y = 43

b x = 61, m = 137

17 a x = 60, y = 37

b x = 65, y = 59

19 x =

c Ellie

11 The whiteboard is longer.

13 a x = 95

18 a x = 103

b Sandy

b x = 137

Height Height (metres) (kilometres) 541.3

0.5413

Shanghai Tower

632

0.632

Petronas Twin Towers

451.9

0.4519

Zifeng Tower

450

0.45

Burj Khalifa

828

0.828

b Burj Khalifa 14 a Desk b The desk is 19 cm longer than the rug.

848

Mathspace WA Curriculum – Year 7 mathspace.co

7 a 27 mm

Reasoning 15 So that there is no confusion as everyone is using the same unit of measurement.

b 38 cm

c 51 cm

d 28 cm

e 30 mm 8 a 48 mm

b 12 m

c 30 mm

16 It is easier to convert by multiplying or dividing by 10, 100, 1000. The Imperial system has inconsistent and difficult conversion factors.

10 48 mm

17 a The distances are too large in space.

11 a 28 cm

b 30 mm

c 48 m

12 a k = 8 m

b n = 2 cm

c x = 21 cm

b Light-years, which is the distance light travels from the sun in one year. Problem-solving 18 495 000 mm, 5.23 m, 499 cm 19 0.19 km, 19 150 cm, 192 m

9 3 mm

13 4 cm 14 a y = 12 m

b n = 23 cm

c d = 14 cm

d k = 6 cm

15 a 4 m

20 2.22 m

16 a 11 cm

21 5.34 m

b 4 cm b 7m

Reasoning

22 356 cm

17 Method 1:

23 0.24 m

P = 2 × (8 + 5)

24 0.74 cm

= 2 × 13 = 26 cm

25 0.7 mm Method 2:

26 60 cm 27 a 1 26 000 cm

=2×8+2×5

b 1.26 km

= 16 + 10 = 26 cm

28 666.67 years Another method:

29 20 years

P=8+8+5+5

30 2 m from each long side and 4 m from the short sides.

12.02 Perimeter

1 a 16 units

b 16 units

2 a 16 units

b 22 units

c 20 units

d 42 units

= 176 cm Fluency b 54 m

d 88 m

e 74 mm

5 a 60 m

c 26 cm b 30 m

b 72 m

c 20 cm

= 26 cm

19 Yes. The side length of the square is 10 cm. If you increase it by 4 times the new side length will be 40 cm, resulting in a new perimeter of 160 cm. 160 cm is 4 times the original perimeter of 40 cm.

3 P = 55 + 73 + 48

4 a 30 m

= 16 + 10 18 No. The side length of the square is 6 cm. If you increase it by 4 cm, the new side length will be 10 cm, resulting in a new perimeter of 40 cm. 40 cm is not 4 times the original perimeter of 24 cm.

Understanding

6 a 28 cm

c 15 mm

d 16 cm

20 Yes. the perimeter will be doubled. Consider a rectangle with a length of 5 units and a width of 3 units. Its perimeter is 2(5) + 2(3) = 16 units. Doubling the length of each side will result to a new perimeter of 2(10) + 2(6) = 32 units which is double the original perimeter. 21 4a cm

Answers mathspace.co

849

22 6p mm

Reasoning

23 10x mm

12 a 60 cm b 60 cm

Problem-solving

c Yes, the answers are the same. The shape can be redrawn so that it becomes a rectangle with sides 17 cm and 13 cm.

24 8 m 25 9 cm

13 P = 5 + 4 + 8 + 5 + 13 + 9

26 70 m

= 44 m or

27 11 cm

P = 2 × (9 + 13)

28 12 cm

= 44 m

29 6 mm

14 Adding sides gives 2 + 6 + 1 + 2 + 5 + 6 = 22 mm. Rectangle method gives 2 × (6 + 5) = 22 mm which is the same.

30 15 m by 5 m 31 38 cm

Problem-solving

32 $4800

15 a 387 m

33 $10 080 34 a 52 cm

b 69 cm

35 a y + 3 + y + 3 + 2x + 2x

c 121 cm b 4x + 2y + 6

36 a 4y – 1 + 3x − 2 + 3x − 2 + 3y − 2 + 3y − 2 b 6x + 10y − 9

b $13 738.50

16 a length = 26 m, width = 16 m b 84 m 17 a 82 m

b 40 m

18 54 m 19 a 144 m

12.03 Perimeter of composite shapes Understanding 1 Add together the length of the sides around the outside of the shape. 2 A and B 3 a 5 sides

b False

4 P = 10 + 7 + 9 + 11 + 9 + 8 P = 54 cm

c 106 m

b $2700

20 a 8.8 m b The trim is added to the existing edges of the tablecloth. So, for each edge, we have 2 metres of trim and another 0.1 metres in each corner. Therefore, the total length of the trim is (2 + 0.1) × 4 = 8.4 m. Since the cost of the trim is $5.40 per metre, the total cost of the trim for the tablecloth is 8.4 m × $5.40 = $45.36.

12.04 Units of area

5 P = 4 + 6 + 3 + 4 + 14 + 4 + 7 + 6 Understanding

P = 48 cm Fluency 6 a 30 cm

b 120 m

c 68 cm

7 48 mm

2 Units of area from smallest to largest: • mm2

8 a 58 m

b 60 mm

• cm2

10 a 54 cm b 58 m

e 18 cm

f 76 cm

c 52 cm

• km2

• Car windscreen

• TV screen

b 36 mm

11 a 58 cm

• m2

3 a Example answers: • Coffee table

9 66 m

850

1 There are 10 mm in 1 cm, 100 cm in 1 m and 1000 m in 1 km.

d 170 m

Mathspace WA Curriculum – Year 7 mathspace.co

b Example answers: • Lid of a shoe box • iPad

Fluency 4 a Square centimetres b Square metres c Square centimetres d Square metres

16 Hectares make it easy to visualise the size of the land. Problem-solving

e Square kilometres

f Square centimetres

g Square metres

h Hectares

i Square metres

j Square metres

18 76 000 m2

k Square kilometres

l Square metres

19 a

17 Approximately 3000 cm2

5 8.41 m

6 Approximately 6912 cm 7 C

8 Approximately 200 mm2 2

Width, m

Area, m2 20 000

200

100

350

7350

100

2400

400

16 000

100

10 000

c Farm 3 and Farm 2

d Farm 5

10 Approximately 26 mm b D

Length, m

b Farm 1 and Farm 4

9 Approximately 130 m

11 a B

Farm

c A

d D

20 a 72 200 m2

b 18.05 acres

12 a 30 000 cm2

b 70 000 cm2

21 a 128 ha

b 1.28 km2

c 560 000 m2

d 60 000 m2

22 a 5.1 ha

b 12.75 acres

f 2 ha

23 46 400 chickens

e 104 000 m Reasoning

13 Area has two dimensions, length and width and 100 × 100 = 10 000 14 a False. The unit of estimation can be different from the unit of measurement used to ﬁnd the exact area, as long as the conversion between units is taken into account. b False. The area of a square is equal to the length of one side squared. c True. The order of the length and width does not matter when calculating the area of a rectangle. d False. This statement is not always true. The area and perimeter are measured in different units (e.g., square units for area and linear units for perimeter), so they aren’t directly comparable in this way. Moreover, for large shapes or shapes with small perimeters but large areas (like a large square), the area can be much larger than the perimeter. e False. When estimating area, it’s important to keep as much accuracy as possible. 15 It can help you check the reasonableness of your exact answer and catch any errors you may have made in your calculations.

24 a 136 ha

b 1 360 000 m2

12.05 Area of rectangles and squares Understanding 1 a 16 square units

b 15 square units

2 cm

3 a i 4 square units

ii 9 square units

iii Shape A b i 25 square units

ii 16 square units

iii Shape B 4 a i 10 square units

ii 9 square units

iii Shape A b i 9 square units

ii 10 square units

iii Shape B c i 16 square units

ii 9 square units

iii Shape A d i 16 square units

ii 25 square units

iii Shape B e i 12 square units

ii 8 square units

iii Shape A f i 18 square units

ii 20 square units

iii Shape B

Answers mathspace.co

851

5 Shape A, Shape C, Shape B

Length

6 Shape A 7 a i Length = 6 units, width = 2 units ii 12 units2 b i Length = 12 units, width = 9 units ii 108 units2 c i Length = 3 units, width = 8 units ii 24 units2 d i Length = 5 units, width = 5 units ii 25 units2

Perimeter

54 m 30 m

50 m

50 m2

c The table from part (b) shows that since area is done through multiplication while perimeter is done through addition and multiplication, having the same area cannot guarantee of having the same perimeter. Problem-solving

23 16 cm2

8 a 24 cm2

b 30 m2

c 50 m2

d 27 mm2

e 138 mm 9 a 81 mm2

b 16 cm2

c 324 m2

10 a i 22 cm

ii 24 cm2

b i 20 cm

ii 24 cm2

c i 14 cm

ii 10 cm2

d i 20 cm

11 a 49 cm2

ii 21 cm b 30 cm2

c 60 cm2

24 $22 500 25 $150 26 150 tiles 27 144 cm2 28 10 boxes

12.06 Area of triangles Understanding

12 5 mm b 64 cm2

13 a 8 cm

1 a

b 25 cm2

c 20 cm2

2 a b = 6 cm, h = 2.7 cm

14 6 mm 15 a 9 m

b b = 36 mm, h = 15 mm

b 2m

c b = 12 cm, h = 6.4 cm

Reasoning

d b = 19.5 cm, h = 12.5 cm

16 a 49 cm2

b 196 cm2

c No

17 Doubling the side lengths does not double the area. 18 a 4 times the original

Fluency 3 a i 50 cm2

ii 25 cm2

ii 15 cm2

ii 11 cm2

b i 30 cm

c i 22 cm

b Each side is multiplied by 2 and the area is found using side × side. As 2 × 2 = 4 the scale factor is 4. c Factor of 25 times the original d Factor of x2 times the original 19 One possible answer is a square with side = 4 units 20 Find the area of the rectangle and halve it.

852

Area

22 24 m2

Fluency

21 a

Width

Length

Width

Area

50 m2

Mathspace WA Curriculum – Year 7 mathspace.co

d i 80 m2 e i 24 cm

ii 40 m2 2

ii 12 cm2

f i 64 m2

ii 32 m2

4 a 21 cm2

b 30 m2

d 1320 cm2

c 24 cm

5 a 36 cm2 2

e 60 cm 6 a Yes

7 b = 8 mm

b 12 m2

c 28 mm2 d 13.5 cm2 2

f 5.2 m b No

c Yes

d Yes

8 a Base = 4 cm

2 Example answer: We breakup the shape into a rectangle and a triangle and add their areas.

b Base = 8 cm

c Base = 2.5 mm

d Base = 20 mm

9 a Height = 9 cm

b Height = 6 cm

c Height = 15 mm

d Height = 10 mm

3 a We breakup the shape into two rectangles and add their areas.

Base (m)

Height (m)

30 2

Area (m2)

b We calculate the area of the large rectangle 6 m × 15 m and subtract the area of the smaller 7 m × 13 m rectangle. Example answer: We can breakup the shape into two rectangles, and add their areas.

Reasoning 1

11 The rectangle can be cut into two equal right triangles. Find the area of the rectangle and halve it to find the area of the triangle.

2m 7m

12 a The area is also halved.

b The area is also doubled.

13 m

13 1 cm × 48 cm, 2 cm × 24 cm, 3 cm × 16 cm, 4 cm × 12 cm 14 No. There are multiple triangles of different dimensions that have the same area. For example, a triangle with the base of 4 cm and height of 10 cm has the same area as a triangle with the base of 8 cm and height of 5 cm, but the triangles are a different shape. b 50 mm2

15 a 16 a

The length of rectangle 1 can be found by adding the two horizontal distances: 6 + 7 = 13 m. And the width of rectangle 1 is 2 m, so the area is 13 × 2 = 26 m2. The length and width of rectangle 2 are 6 m and 13 m, so the area is 6 × 13 = 78 m2. So by adding the areas of the two shapes, the area of the composite shapes is 104 m2. Fluency

Problem-solving

4 a 56 cm2

17 48 m2

5 56 m2

18 150 m2 19 a 2000 cm 20 a 300 m2

6 a 6 cm by 3 cm 2

b 0.2 m

b $7500

7 a i 68 cm2

24 ℎ = 8 cm

12.07 Area of composite shapes

ii 8 cm by 6 cm iv 116 cm2

ii 4 mm by 4 mm

iv 40 mm2

iii 16 mm b 146 tiles

b 18 cm2

b i 24 mm

22 5 litres

c 280 cm2

d 60 cm2

c 42 cm

iii 48 cm

21 $600

23 a 132 tiles

b 224 cm2

8 a 96 cm2

b 112 m2

c 78 mm

d 143 cm2

e 37 cm2

f 120 cm2

g 70 cm Understanding 1 a Rectangle and parallelogram b Example answer: Add the area of the rectangle and parallelogram together.

9 a 117 cm2 b 15 cm2

c 102 cm2

10 a 264 m2 b 136 cm2 c 80 m2

d 120 cm2

Answers mathspace.co

853

9 a 84 ha

Reasoning 11 a 5 × 9 + 5 × 8 = 85 m2

10 54 cm2

b 9 × 13 − 8 × 4 = 85 m

11 180 m2

12 35 × 39 − 9 × 11 = 1266 m2. This is the most efficient method because it only involves two rectangle area calculations. If we used addition we would need to calculate three rectangle areas.

12 $750 13 $1100 14 $742.00 15 a 143 m2

13 Find the area of the interior rectangle shaded in figure 2.

13.01 Nets of solids

Find the area of the corner rectangle shaded in figure 3.

Understanding

Find the area of the entire rectangle shaded in figure 4.

1 a Triangular prism c Octagonal pyramid

Calculate the area of the shaded region in figure 1 by subtracting the interior rectangle and two times the corner rectangle from the entire rectangle of figure 4.

Fluency

Problem-solving

3 a Triangular prism

14 a 70 m

15 264 cm2 16 a 414 m2

b 96 m2

c 318 m2

d $8268 b 184 m2

b $1716

b Rectangular prism d Octagonal prism

2 A net of a solid is a 2-dimensional representation of a 3-dimensional solid.

b Rectangular pyramid

c Pentagonal pyramid d Octagonal prism

b $700

17 a 416 m2

b 840 000 m2

4 a

c $2308.80

18 $4481.25 2

19 450 cm

20 a 109 m2

b $1308

21 a 22 603 m2

b 33 904.5 kg

Chapter 12 review 1 a 88 cm2

b 49 mm2

2 a 8m

b 11 m 2

3 a 20 cm

b 42 cm2

4 a Base = 8 cm

b Base = 10 cm

c Base = 6 mm

d Base = 12 mm

5 a 135 cm2

b 99 cm2 2

6 a 116.5 cm

b 89 m

7 Approximately 156 mm 8 66 000 m2

854

Mathspace WA Curriculum – Year 7 mathspace.co

5 a Square pyramid

b Pentagonal pyramid

c Hexagonal pyramid d Octagonal prism 6 a Triangular prism c Pentagonal pyramid 7 Rectangle

b Square prism

8 Example answers:

b i

• Rectangular pyramid • Rectangular prism • Triangular prism

iii

Reasoning 9 In order to find the surface area of a solid when using its net, calculate the area of each 2D face/base and then add up each area to get the total surface area. 10 A prism and a pyramid can be classified by their base, and the number of faces a prism or a pyramid will have is the same as the number of sides of its base. A prism will have two bases and a pyramid will only have one base. 11 The sketch and reasoning of the net should look proportionate to the given length, width, and height of the rectangular prism described.

c i

iii

d i

iii

Problem-solving 12 a Answers will vary.

b Answers will vary.

c Answers will vary.

d Answers will vary.

13 Answer may look like:

e i

The number of faces in a net is equal to the number of faces in its corresponding 3-dimensional solid. The number of edges and vertices may be different due to the folding and joining of the faces

iii

13.02 Visualise solids Understanding 1 A 2D drawing of a 3D shape. 2 A perspective drawing is more realistic. Fluency 3 a i

4 a D

b C

c B

5 A 6 a Cube

b Rectangular prism

c Triangular prism

d Cylinder

e Hexagonal prism

f Pentagonal prism

7 a Hexagonal prism or pyramid b Trapezoidal prism or pyramid iii

c Cylinder or cone or sphere 8 A 9 B 10 B 11 D

Answers mathspace.co

855

12 a i

iii

b i

g i

iii

h i

iii

c i

iii

i i

iii

iii d i

j i

iii iii e i

Reasoning iii f i

iii

856

Mathspace WA Curriculum – Year 7 mathspace.co

13 D. By process of elimination, we can see that option A has a hexagon as its top view while options B and C each have two rectangles as its top view. The only solid with three rectangles would be option D. 14 Yashi’s observation is accurate. Both the first and second solids share an identical ‘U’-shaped front view. However, their side views differ, as the first solid comprises just three squares, in contrast to the five distinct squares forming the second solid. Moreover, their top views are not the same; the first solid is composed of four squares, while the second solid features six squares.

15 The front view has a hole through the middle that is missed by the side view, and the top view has cubes cut out such that they would be missed by the side view.

b 9 rectangular prisms 1 × 1 × 48 1 × 2 × 24 1 × 3 × 16

Problem-solving

1 × 4 × 12

16 Answers will vary.

1×6×8 2 × 2 × 12

17 Answers will vary.

2×3×8

13.03 Volume using unit cubes

2×4×6 3×4×4

Understanding 1 a It is the amount of space that is contained within that shape. b A cube with side length 1 unit. 2 a One possible solution:

c 12 rectangular prisms 1 × 1 × 72 1 × 2 × 36 1 × 3 × 24 1 × 4 × 18 1 × 6 × 12 1×8×9

b One possible solution:

2 × 2 × 18 2 × 3 × 12 2×4×9

c One possible solution:

2×6×6 3×3×8 3×4×6 d 16 rectangular prisms

d One possible solution:

1 × 1 × 120 1 × 2 × 60 1 × 3 × 40 1 × 4 × 30

3 a 18 unit cubes

b 16 unit cubes

c 36 unit cubes

d 99 unit cubes

e 140 unit cubes

1 × 5 × 24 1 × 6 × 20 1 × 8 × 15 1 × 10 × 12

4 a 4 rectangular prisms

2 × 2 × 30

1 × 1 × 12

2 × 3 × 20

1×2×6

2 × 4 × 15

1×3×4

2 × 5 × 12

2×2×3

2 × 6 × 10 3 × 4 × 10 3×5×8 4×5×6

Answers mathspace.co

857

Fluency 5 a 4 cm

e 10 cm3

b 9 cm

c 8 cm

d 45 cm

f 20 cm3 g 27 cm3

i 48 cm3 Reasoning 6 No, the volume will not change because no unit cubes are added or taken away from the rectangular prism simply by turning it. 7 Since 1 metre is 100 centimetres, and the edge of a 1-centimetre cube is 1 centimetre, we can stack 100 layers of a square area that is 100 cubes long and wide. 8 We know that if we lay one row of unit cubes the length and width of the rectangular prism, we can calculate the area of the top of the layer using the area formula, A = l × w. Then, we can stack how ever many layers of the unit cubes that are needed to reach the height of the rectangular prism. The area of the first layer multiplied by the number of layers in the rectangular prism is equal to the volume of the rectangular prism, l × w × h. Problem-solving 9 98 cubic units

13.04 Units of volume Understanding 1 Answer may look like: Volume is a measure of how much space an object occupies for example how much space bottles take on a supermarket shelf. Capacity is a specific type of volume, it measures the amount of liquid or gas that a container can hold, such as how much liquid is in a bottle. 2 a Answers vary. One possible solution: A small seed. b Answers vary. One possible solution: A sugar cube. c Answers vary. One possible solution: A washing machine. d Answers vary. One possible solution: A drop of liquid from an eyedropper. e Answers vary. One possible solution: A bottle of soda.

858

Mathspace WA Curriculum – Year 7 mathspace.co

f A nswers vary. One possible solution: A large shipping container. g Answers vary. One possible solution: A lake. Fluency 3 a No

b No

c Yes

c cm3

b m

4 a m

d m3

5 Litres and millilitres 6 mL 7 mL 8 C 9 a cm3

b Approximately 56 cm3

10 Approximately 44 250 cm3 11 Approximately 2.7 L 12 Approximately 1.2 L Reasoning 13 1200 mL. The liquid fills half of the container, so half of the total amount of liquid that could fill it is the amount in the container. 14 a One possible answer: The books on the shelves could be distributed to fill two of the shelves, so we could multiply the volume by so that the estimated volume of the books is 0.32 m3. b One possible answer: Looking at the books on the shelves, it seems that the books could take up about of one shelf, so multiplying the volume by , the books may have a volume of 0.0267 m3. Problem-solving 15 200 days. 16 1200 mL. 17 a The 2 L bottle. b Buying 2 bottles of 500 mL and a 2 L soda for a total of $12.80.

13.05 Volume of right prisms Understanding 1 One possible answer: To determine the volume of a composite solid, you need to calculate the volume of each individual simple shape that forms the composite solid, and subsequently sum up these volumes. 2 One reason that every cube is a rectangular prism and not every rectangular prism is a cube follows: we can use the formula for the volume of a rectangular prism when calculating the volume of any cube, but we cannot use the formula for the volume of a cube when calculating the volume of any rectangular prism.

b 2250 cm3 3

c 645.975 m

d 288 m3

e 32 cm3

f 88 mm3

4 a 1728 cm3

b 216 cm3

5 2744 m3

7 90 000 cm3 8 a 4900 cm3 b i 16 537.5 cm3

it by

Then, multiply the length, width and height of the box to find the cubic centimetres: 200 cm × 30 cm × 40 cm = 240 000 cm3.

Problem-solving

ii No

17 288 000 L

9 68 600 cm3

18 a 37.5 litres.

10 45 000 cm3 11 a 252 cm3

b 588 cm3

c 840 cm3

13 a 1344 units

15 First, convert 2 m to centimetres by multiplying

b We could calculate the volume of the solid by breaking it into four rectangular prisms and finding the sum of their volumes.

6 3 cm

12 a 168 cm

We could argue that calculating the volume of one rectangular prism and recognizing that four prism make up the composite solid leads to more efficient calculation.

16 a In order to calculate the volume of the solid, we need to calculate the volume of the entire solid with the outermost dimensions and the volume of the rectangular prism that was cut out. Then, we will subtract the volume of the rectangular prism that has been cut out from the volume of the entire solid. The volume of the solid shown is 1080 cm3.

Fluency 3 a 720 mm3

Another way to calculate the volume is by breaking the solid into two rectangular prisms: the smaller one on top and the larger 21 cm × 16 cm × 7 cm prism. The volume of each prism is 784 cm3 and 2352 cm3, respectively. The sum of the volumes is the volume of the composite solid.

b 112 cm 3

c 68 cm

c 280 cm

b 88 cm3 d 384 cm3

Reasoning 14 The volume is 3136 cm3. One way to calculate the volume is by breaking the composite solid into four identical rectangular prisms. Then, one of the rectangular prisms is 16 cm × 7 cm × 7 cm = 784 cm3 so the four that make up the composite solid is four times that volume.

b One possible answer: The aquarium should not be filled to the top in case of spilling over, so we could assume that the water could be filled with 2 centimetres of space from the surface of the water to the top of the tank. Then the volume of the aquarium is 35 000 cm3, meaning that a reasonable amount of water to fill the aquarium is 35 litres. 19 10 units 20 7.81 cm 21 2.72 m3

Answers mathspace.co

859

13.06 Attributes in measurement formulas

Understanding

e Volume (cm)3

1 a Constant

b Varying

c Varying

d Constant

2 a Distance

b 120

c Time

d Speed

3 a Base, height, area

80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5

b Base

c 12

d Height and area

e None

f A=6×h

Width (cm)

8 a 30 cm3 b

Length 4 4 4

Width 1 2 3 Area

9 10

f V = 8h

9 10

Height

Volume 40 50 60 70 80 90

100

d Linear

9 a

Time (hours)

d V = 350 cm 1

9 10

Speed 360 180 120 90 72 60 45 40 36 (km/h)

30 25

varying width, length = 4 cm

Speed (km/h) 360

300

Width 1 2 3 4 5 6 7 8 9 10

240 180

Length

9 10

120

Area

6 12 18 24 30 36 42 48 54 60

b A = 6l c Area 60

d Linear

c No

Time (hours) 1 2 3 4 5 6 7 8 9 10

d 24 km/hr

Reasoning varying length, width = 6 cm

Length 1 2 3 4 5 6 7 8 9 10

7 a 8 m3

860

Width

c V = 10h

Length

4 8 12 16 20 24 28 32 36 40

c Area

55 50 45 40 35 30 25 20 15 10 5

b A = 4w

6 a

Lenght (cm) 1 2 3 4 5 6 7 8 9 10

Length (cm)

5 a

Volume 8 16 24 32 40 48 56 64 72 80

Fluency 4

Height

b 16 m3

c 24 m3

Mathspace WA Curriculum – Year 7 mathspace.co

10 Constant means “always the same” and a constant term never changes its value. Variable means “changeable” and a variable term such as w is a term that can take on many different values. 11 The sides of a square are equal. Therefore, the length must be a constant of 12 cm.

14 792 m3

Problem-solving 12 a True

b True

c False

d False

15 7 units

13 a True

b False

c True

d True

16 7.5 cm

14 a True

b False

c True

d False

17 a Constant

b Varying

c Varying

d Constant

15 Double the base

18 a Length 6

16 Double his speed

Chapter 13 review 1 a Triangular pyramid

b Rectangular prism

c Pentagonal pyramid d Pentagonal prism 2 a Square pyramid

9 10

Area

6 12 18 24 30 36 42 48 54 60

b A = 6w d Linear

c Area 40

b Pentagonal pyramid

c Hexagonal pyramid d Octagonal prism 3 a

Width

30 25

varying width, length = 6 cm

20 15 10 5

Width 1 2 3 4 5 6 7 8 9 10

19 a 90 cm3 b

4 a i

iii

Width

Length

Height

Volume 108 126 144 162 180 198 216 d V = 450cm3

c V = 18h b i

iii

5 a 4

b 6

6 a m3

c cm3

b m3

d cm3

7 a 280 mm3

b 210 cm3

c 2145 mm 8 43 750 cm

d 280 cm

9 a 4807 cm3 b i 16 223.625 cm3 10 a 32 cm3 c 150 mm3

ii No b 304.75 cm3 d 432 cm3

11 k = 11.5

c 1024 cm3

b True

c False

d False

14.01 Transformation of shapes Understanding 1 A transformation is when an object or point is moved, turned, flipped, or changed in shape or size. 2 a Yes

b No

c No

d No

3 a It slides the shape. b It turns the shape over. c It spins the shape. Fluency

12 y = 10.1 13 a 76 cm3

20 a True

b 184 cm3 d 452 cm3

4 a No

b No

e No

f Yes

c Yes

d No

Answers mathspace.co

861

5 a No

b No

c Yes

d No

6 a Yes

b No

c No

d No

e No

f Yes

e Yes

Fluency 5 a 8 units 6 (−3, 5) 7 a

7 a Translation

b Rotation

c Translation

d Reflection

e Translation

f Reflection

8 a Translation

b Rotation

c Reflection

d Reflection

10 We cannot use reflection because flag C is not a mirror image of flag A.

13 a Cart C is rotated. b Example answer: To move cart A to cart C using translation, adjust the track to form a complete circle or loop.

b Translate the apple

15 Rotation 16 Example answer: Translation up and then to the left

14.02 Translations Understanding 1 a Yes

b No

c Yes

2 When objects are translated, the size of the image remains unchanged. 3 When a point is translated, the values of both the x and y-coordinates might change. 4 a Up

862

b Left

c Down

d Right

Mathspace WA Curriculum – Year 7 mathspace.co

5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Problem-solving 14 a Rotate the left arm

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

11 No. Block E and block N have the same vertical distance to the top of the tower, but different horizontal distances. 12 Figures E and G because they are the same shape and area.

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Reasoning 9 Reflection and translation. Flag B is a mirror image of flag A but on a higher position. This means flag B is a transformation of flag A using reflection and translation.

b 3 units

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

x 1 2 3 4 5

5 4 3 2 1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

7 6 B

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

3 A

2 1

x 1

0 y

x 1 2 3 4 5

3 2 1

x 2 3 4 5 6 7 8 9

e y

10 9 8 7 6 5 4 3 2 1

8 6

x 6

y A

B x 1

x 1

9 (1, 1)

10 a

x 1

Answers mathspace.co

863

Problem-solving

14 a 3 units up

b 5 units left

c 3 units right and 4 units down

d 5 units right and 8 units up

5 4

15 a 6 units left

b 9 units left and 5 units up

2 1

1 2 3 4 5 6 7 8 9 10

16 a 7 units up b 9 units left and 4 units down 17 a 5 units right and 3 units up

y 8

b 4 units right and 1 unit down

c 5 units up

d 1 unit right and 1 unit down

14.03 Reflections

4 A

Understanding

2 B

1 0

x 5

2 C

y 8 7 A

3 a Yes

b Yes

c No

d Yes

4 a Yes

b Yes

c No

d Yes

Fluency

1 A

5 a

3 2 1

7 6

4 3 2 1 0

x 1

2 3 4 5 6 7 8 9

Reasoning 12 B or D 13 a 3 units right and 1 unit down b 1 unit right and 2 units up c 2 units left and 4 units down d 6 units right and 7 units up

864

Mathspace WA Curriculum – Year 7 mathspace.co

6 a

4 2

−5 −4 −3 −2 −1 −2

1 2 3 4 5

−4 −6 x

−8 −10

7 a

y 4

2 x −8 −6 −4 −2

−2 −4

Answers mathspace.co

865

7 y 6 5 4 3 2 1 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

6 y 5 4 3 2 1

−8 −6 −4 −2−1

1 2 3 4 5 6

13 a

10 8 6 4 2

6 4 2 −4 −2 −2

x 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

−4 −6

12 a

5 4 3 2 1 −1 −1 −2 −3 −4 −5

b A′

1 2 3 4 5 6 7 8 9

A′

10 8 6 4 2

−10−8 −6 −4 −2 −2 −4 A −6 −8 −10

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 A −3 −4 −5

866

10 8 6 4 2

−2 −3 −4 −5 −6

A′

x 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

A y

x 2 4 6 8 10

Reasoning

x 2 4 6 8 10

14 a i a = 9

ii b = 37

iv d = 8

v e = 150

b On itself 15 a True

b False

c True

e True Problem-solving 16 (3, −4)

17 A′(−2, −1), B (2, −4) and C′(4, −2)

x 1 2 3 4 5 A′

Mathspace WA Curriculum – Year 7 mathspace.co

iii c = 8

d False

18 Point C. Point B is a reflection of point H across the y-axis, which serves as the line of reflection. To minimise the distance you and your friend will have to walk, you should park your vehicle at the midpoint between points B and H, which lies only on the y-axis. Point C greatly minimises the distance and is the correct answer since it is inside the blue rectangle and closest to the points.

y 5 4 3

1 x

Understanding

y 5

1 a No

b Yes

c No

d No

2 a 270°

b 180°

c 90°

d 315°

3 B

Fluency

4 (4, −4)

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Reasoning 9 Yes. Example answer: x 1 2 3 4 5

A square can be rotated by 90°, 180°, or 270° using the intersection of the diagonals as the centre of rotation and it will still look exactly the same.

6 B 7 Triangle P y

10 a Yes, Aneta rotated the segment clockwise around the origin instead of anticlockwise.

5 4

b A′ = (−4, 3) and B′ = (−1, 2)

11 a Yes, when rotated 180° each face is showing the dots in the same arrangement as before they were rotated.

2 1

x 0

14.04 Rotations

8 a

b No, only the faces with one/four/five dot(s) gives the exact same face when they are rotated 90°.

Answers mathspace.co

867

if we apply two translations, the result will be the same regardless of the order of the translations. Similarly, applying two rotations around the same center will yield the same result regardless of the order in which the rotations are applied. In these specific cases, the final image will be the same regardless of the order of transformations.

Problem-solving 12 13 y 12 11 10 9 8 7 6 5 4 3 2 1

Fluency x 1 2 3 4 5 6 7 8 9 10 11 12 13

13 a i 270°

ii 90°

b i 180°

ii 180°

14 a Segment JK

b Segment PQ

15 a 90° anticlockwise

b 180° clockwise

c 90° clockwise

d 270° clockwise

16 Avery needs to turn the map 180° clockwise or anticlockwise. He is currently facing south, so to have south pointing up on the map it needs to be turned upside down.

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

4 a

14.05 Multiple transformations Understanding 1 Here are three main types of transformations:

• Rotation: This is when you turn an object around a fixed point. You can turn it a little bit or a lot, clockwise or anticlockwise. 2 The order of transformations can change the image of a shape. Even if the same transformations are used, if they are applied in different orders, the images will be different. For example, if we rotate a square first and then move it, we will get a different shape than if we move it first and then rotate it. The order matters because some transformations do not work together in the same way, so the way we put them together can make a big difference. However, there are cases where the order of transformations does not matter. For instance,

868

Mathspace WA Curriculum – Year 7 mathspace.co

x 1 2 3 4 5

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 A′ −4 −5

• Translation: This is when you move an object in a straight line without turning it. You can move it up, down, left, right, or diagonally. • Reflection: This is when you flip an object over a line of symmetry. You can flip it horizontally, vertically, or diagonally.

5 4 3 2 1

1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 a

5 4 3 2 1 −5 −4 −3 −2 −1 A′ −1 −2 −3 −4 −5

1 2 3 4 5

A′′

x 1 2 3 4 5

A′′

5 4 3 2 1

14 A″ (−6, −8) 15 a B (10, −6) x

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

6 a

5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

A′

−5 −4 −3 −2 −1 −1 A′′ −2 −3 −4 −5

x 1 2 3 4 5

5 4 3 2 1

x 1 2 3 4 5

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

8 (6, 9) 9 (1, 2) 10 a (−5, 6)

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

7 a A′(−3, 4) 5 4 3 2 1

5 4 3 2 1

x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5 A′′

b 20 units

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

13 A″ (5, 4)

1 2 3 4 5

x 1 2 3 4 5

b (6, −5)

11 (−8, −2) 12 (3, 6)

Answers mathspace.co

869

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

d i

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

y A′

x 1 2 3 4 5

ii Translation 1 unit right and 3 units up. Reasoning 21 a i

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

22 a Rotate section a1 180° clockwise or anticlockwise about the centre of the flag.

b Reflect section a1 over the horizontal midline, then the vertical midline (in either order). x A′ 1 2 3 4 5

ii Translation of 1 unit left and 1 unit down. b i

5 4 3 2 1

b Rotation and translation 25 a Reflection and translation x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

27 Triangle W 28 Square B

14.06 Line and rotational symmetry Understanding 1 a Line c Rotational

x 1 2 3 4 5

A′

ii Translation 4 units right and 6 units down.

870

b Rotation and translation 26 Shape A

ii Translation 5 units left and 1 unit down. 5 4 3 2 1

Problem-solving 24 a Reflection and translation

A′ −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

c i

23 Answers may vary. Starting with the top leftmost parallelogram in the pattern, perform a series of translations down the fabric until you have six parallelograms. Then, reflect the set of six parallelograms across the line of symmetry in the pattern.

Mathspace WA Curriculum – Year 7 mathspace.co

b Line d Rotational

e Rotational 2 Not all shapes have line symmetry. For example, a rectangle has line symmetry, but a scalene triangle does not. A rectangle can be divided into two identical halves by a vertical line through its centre, while a scalene triangle cannot be divided into two identical halves by a line.

3 Not all shapes have rotational symmetry. For example, a square has rotational symmetry, but a scalene triangle does not. A square can be rotated by 90° and still look the same, while a scalene triangle will look different when rotated by 90°. Fluency 4 a 3

b 0

c 2

e 1

f 1

g 1

d 5

5 0, 3, 8 6 a No

b Yes

c Yes

d Yes

7 a No

b No

c Yes

d No

e Yes

f No

g No

h No

i No

j No

k Yes

l Yes

m No

n Yes

o No

p Yes

8 a Yes

b No

9 a Line

b Line and Rotational

c Neither 10 a

11 AE and CG

17 No. All isosceles triangles have line symmetry about their midline. However, an isosceles triangle can have rotational symmetry only when it is an equilateral triangle. 18 Yes. An equilateral triangle is a triangle where all three sides are of equal length. It can be rotated by any multiple of 120° and still look the same as it did before the rotation. 19 Yes. In a regular polygon, all sides and angles are congruent, so when the shape is rotated it will look exactly the same as it did before the rotation. To find the angle of rotation, we can use the equation Angle of Rotation = where n is the order of rotation.

20 a Yes, the given shape has rotational symmetry. The shape is centered the origin and its vertices are (−3, −3), (−3, 3), (3, 3), (3, −3). If you rotate the shape 90°, 180°, or 270° around the origin, each vertex moves to another vertex’s original position. For instance, if you rotate the shape 90° around the origin, (−3, −3) will move to −3, 3, (−3, 3) will move to (3, 3), (3, 3) will move to (3, −3) and (3, −3) will move to (−3, −3). This rotational symmetry around the origin confirms that the shape has rotational symmetry. b No, because based on the coordinates (−2, 1), (−1, −2) and (2, 3), after the rotation, the new coordinates would be (2, −1), (1, 2), and (2, 3). Since these new points do not match the original positions, the shape does not exhibit rotational symmetry.

12 EG and HF 13 a

Reasoning

Problem-solving 21 a A, B, C, D, E, H, I, K, M, O, T, U, V, W, X, Y 14 a 3

b H, I, O, X, o, x

b No

15 a 2

b 2

c 4

d 4

22 a 2

e 5

f 3

g 5

h 4

23 When the fan is rotated by 90°, 180° and 270°, the fan is mapped on to its original position.

16 a 180°

b 90°

24 a 180 cm

b 120 cm

b 35 times

Answers mathspace.co

871

Chapter 14 review 1 a Yes

y 8

b No

c No

d No

6 4

2 When a point is translated, the values of both the x and y-coordinates might change. 3 a Left

b Right and down

c Right and up

b (−4, 2)

c (3, 4)

d (3, −2)

−8 −6 −4 −2 −2

d Right

4 a (3, 4)

−4 −6

11 a

−8 −6 −4 −2 −2

4 3

−4

−6

1 0

7 5

x 1

2 3 4 5 6 7

b X (−5, 1)

7 y 6 5 4 3 2 1 −7 −6 −5 −4−3−2−1−1 −2 −3 −4 −5 −6 −7

−8

8 9

6 a A (1, 1)

c M (1, 4)

b B (−8, 5) 12 a 285°

14 a x

1 2 3 4 5 6 7

4 3 P

2 1

x 0

y 5

4 2 −5 −4 −3 −2 −1 −2

c 255°

b Rotation of 180° around the origin. 6

b 180°

13 (0, −2)

8 a 3 units right and 2 units down

2 4 6 8 10

x 1 2 3 4 5

−4 −6 −8

4 3 P

2 1

−10 0

872

Mathspace WA Curriculum – Year 7 mathspace.co

d 300°

15 13 y 12 11 10 9 8 7 6 5 4 3 2 1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5 6 7 8 9 10 11 12 13

16 270°

x 1 2 3 4 5

20 (9, 10)

17 a

5 4 3 2 1

22 (11, 7) x

−5 −4 −3 −2 −1 −1 −2 −3 A′ −4 −5

5 4 3 2 1

A″

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

21 (1, 3)

1 2 3 4 5

23 a Yes

b No

c Yes

d Yes

e Yes

f Yes

g Yes

h Yes

24 For rotational symmetry the angle of rotation is the angle by which a shape can be rotated about its centre and still look the same. The order of rotational symmetry indicates the number of times a shape can be rotated within a full 360-degree circle and still look the same.

For the given orders of rotational symmetry: Order 2: x 1 2 3 4 5

. The shape can be rotated

by 180° and still look the same. An example of this is a rectangle (excluding a square) that has two axes of symmetry. Order 3:

. The shape can be rotated

by 120° and still look the same. An equilateral 18

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

triangle has three axes of symmetry and an order of 3 rotational symmetry. Order 4: x 1 2 3 4 5

. The shape can be rotated

by 90° and still look the same. A square has four axes of symmetry and an order of 4 rotational symmetry. Order 5:

. The shape can be rotated

by 72° and still look the same. A regular pentagon has five axes of symmetry and an order of 5 rotational symmetry.

Answers mathspace.co

873

To generalise the angle of rotation for any order of rotational symmetry, you can use the following formula: Angle of Rotation = where “Order” represents the order of rotational symmetry. This formula calculates the angle by which a shape can be rotated and still look the same for any given order of rotational symmetry.

15.01 Language of probability

b An outcome

2 Sample space 3 a True

b True

18 Unlikely. Since there are 12 sides on the die, there are 12 outcomes total. Rolling an 8 or higher has 5 outcomes: 8, 9, 10, 11, 12. Since 5 outcomes is less than half the total number of outcomes, rolling an 8 or higher is unlikely. 19 C The prime numbers between 1 and 6 are 2, 3, and 5. Rolling a prime number has an even chance. The even numbers on a standard die are 2, 4, and 6, so rolling an even number also has an even chance.

Understanding 1 a A trial

Reasoning

c Not true d True

The numbers less than 3 are 1 and 2. There are 6 total outcomes when rolling a die, so 2 outcomes is less than half, which means rolling a number less than 3 is unlikely.

Fluency

20 The order is 2, 3, 1, 4.

4 a A, B, C

1. Half the cards are black, so this has an even chance.

b 1, 2, 3, 4, 5, 6

5 6, 7, 8, 9, 10, 11, 12 6 a Unlikely

2. There are only 4 sevens, so this is unlikely. b Impossible

c Even chance

d Impossible

e Unlikely

f Certain

7 a Impossible

b Even chance

c Certain 8 a Unlikely

b Even chance

c Unlikely

d Impossible

9 a Likely

b Impossible

10 a No

b No

c Yes

11 a Apple

b Unlikely c C

d D

12 a Yes

b Yes

d No

c No

3. There are 13 diamonds, so this is unlikely, but more likely than drawing a seven. 4. There are 9 numbers from 2–10, and there are 4 of each number, so there are 9 × 4 = 36 cards numbered 2–10. This is more than half the deck, so this is likely. 21 An example is a bag containing 3 blue marbles, 3 red marbles, and 3 green marbles. Drawing any colour is equally likely as the next since there are 3 of each colour, but the colours do not have an even chance of being drawn. Even chance refers to one event having half the total number of outcomes. Equally likely means all the events have the same number of outcomes.

13 a Green

b Blue

14 a Blue

b Likely

15 a Impossible

b Likely

16 a Even chance

b Unlikely

22 Marie is not correct. Even if there are fewer Fords than Holdens, there is still a chance for the Fords to win. It can be said that Ford is unlikely to win but not impossible to win.

17 a Certain

b Even chance

Problem-solving

c Impossible

d Unlikely

e Certain

23 a True 24 Lisa

874

Mathspace WA Curriculum – Year 7 mathspace.co

b False

c True

d False

25 a 4 b Yes, there are 13 of each suit, so each suit has an equal chance of being drawn. c Drawing red or drawing black

15 If 4 marbles are black, then 23 – 4 = 19 marbles are not black. This means there is a probability of

15.02 Probability as a number

that Paul will pick a marble that is not

black. 16 No, the bag only contains red, blue, green, and orange marbles. This is because the probabilities of the colours listed sum to 100%, which means these colours represent all the marbles in the bag.

Understanding 1 Probability = 2 Probability =

17 Spinning a red or a yellow

Fluency 3 a Unlikely

There are 8 total sections on the spinner. 2 are red and 2 are yellow, so each have a probability

b Certain

c Impossible

d Likely

e Even chance

f Likely

of occurring, they are equally likely.

g Unlikely

h Unlikely

i Likely

j Likely

k Unlikely

l Even chance

4 a

. Since they have the same probability

Problem-solving 18 a

d 19 Here is an example answer:

5 a

d 0

6 a

b 0

b 75%

c 25%

d 50%

A spinner with four colours, where red and blue are likely to occur, and yellow and green are unlikely to occur.

e 7 a 100% 8

20 a

21 a

10 a 35%

b 60% b

15.03 Theoretical probability d

Understanding 1 7 2 a True

12 a 27, 28, 29, 30, 31, 32, 33, 34 b 13 a

9 0.8

11 a

c False

d True

Fluency

c b

b False

d 10

3 4 a

Reasoning 5 a 822, 827, 828, 872, 877, 878, 882, 887, 888 14 a 4

b 60%

Answers mathspace.co

875

6 12

15.04 Relative frequency

7 3

Understanding

8 a 6

b 3

9 a

1 Relative frequency =

Reasoning

2 Relative frequency is the probability of an event based on actual observations or experiments, while theoretical probability is the probability of an event based on the mathematical likelihood of the outcomes.

12 4 times.

3 a Relative frequency

10 a

11 35

To find the expected number of outcomes, we multiply the probability of rolling a 1 by the number of trials. The probability of rolling a 1 on × 24 = 4.

a standard die is , and

b Relative frequency c Theoretical probability Fluency 4 a 90

6 a

7 a

13 Example answer: No, probability helps us predict outcomes, but it does not guarantee that the outcomes will happen. Although it is expected that the coin will land on heads 5 out of 10 times, there is still a chance that it will not land on heads exactly 5 times. 14 a Lose the game b Yes, it is possible because there is still a

5 6%

8 a 157 9 a 110 b i

chance of rolling a 1. The probability is unlikely, but not impossible.

b No

b 50

c 155

d 300

18 a Twenty-sided die

b Four-sided die

20 a 0

b 2

876

11 1265 12 a 7

13 a 5

c 60

d 5

Mathspace WA Curriculum – Year 7 mathspace.co

Reasoning 16 a

c Ten-sided die b 130

b 270

15 a 4

17 a 0

19 a 65

iii

Problem-solving 16 a 145

10 a

c No, Elsie is only expected to win money 1 out of the 3 times. However, she will need to pay a total of $3 to play those 3 times. Even if she won $2 once, it would not make up for the $3 she paid to play. 15 a

b 20%

b 10

d No, because an experiment does not always go the way we expect. 17 A deck of cards has the same number of red and black cards. A deck of cards also has the same number of cards in each suit. Table 1 shows almost the same number of red and

black cards while Table 2 shows same number of Spade and Diamond cards, but there is a big difference from the numbers of Heart and Club cards. This means that Table 1 has the closest results to the expected outcome. 18 a Theoretically, each number should occur once. Experimentally, we see the numbers occurred once, twice, or no times. Since these are close to the theoretical probabilities, the program appears to be fair. b Theoretically, each number should occur 10 times. Experimentally, we see each of the numbers occurred close to 10 times. Since these are close to the theoretical probabilities, the program still appears to be fair.

b Tuesday

20 a

21 a 200

c 85%

Cricket

Softball

9 a 30

b 6

10 a 169

b 223

c 192

e 262

f 293

11 a i 51

ii 40

iii 10

iii

b i

d 20

8 a 9

Problem-solving 19 a 76%

7 Skateboard Bike

12 a

d 138

13 a

14 a

15 a

16 a 55

15.05 Venn diagrams

Understanding 1 a Multiples of 6 and multiples of 8. b 6, 12, 18, 24, 30, 36 c 8, 16, 24, 32, 40 d 24 2 a Students who have both Sumsang and jPhones. b Students who have a Sumsang or jPhone or both. c Students who have neither Sumsang or jPhones. d The total number of students or universal set.

17 a 18

b 6 Cake 19

Ice cream 12

Fluency 3 a 3

b 1

c 4

4 a

5 a 20

b 14

c 17

6 a 6

b 2

c 6

Reasoning 19 a 7 students sat for both tests. Since 27 students sat for the original and 10 sat for the make-up test, there were a total of 27 + 10 = 37 students who sat for both tests. Since there are only 30 students in the class, 37 − 30 = 7 students must have sat for both.

Answers mathspace.co

877

24 a

Original test Catch-up test

Tennis 20

Hockey

4 3 4

5 5

20 Eiichiro is correct in saying that 70 people like watching basketball in total, and that 63 people like watching baseball in total. Adding 70 + 63 + 31 to get 164, however, is incorrect as this includes the 49 people who enjoy watching both sports twice. To correctly calculate the total number of people, we can just add the four numbers shown in the regions of the Venn diagram: 31 + 21 + 49 + 14 = 115 people. (This is also the result that we get by subtracting the extra 49 from Eiichiro’s calculation of 164.) 21 One student studies German, so we should add 1 into the top-left circle. Since the student studies no other languages, we should add it to the part of the circle which doesn’t overlap the other circles. So we should increase the 16 to a 17. One student studies both Italian and German, so we should increase the overlap section between German and Italian by 1, from 4 to 5. Similarly, one student studies Spanish and Italian, so we should increase this overlap section from 9 to 10. The last two students don’t study any languages, so they don’t belong in any of the circles and we should increase the 0 in the outer rectangle to be 2 instead. 2 German

Italian

13 2 12

10 36 Spanish

Problem-solving 22 Prime numbers 23 26

878

Mathspace WA Curriculum – Year 7 mathspace.co

Cricket

b 25 a Answers may vary. Example diagram: Trapezium One pair of parallel sides

Parallelogram Two pairs of parallel sides

Rhombus All sides equal

Kite Two pairs of adjacent equal sides

(Opposite sides are equal) Rectangle All angles right

Square

b Answers may look like: 1. Which quadrilaterals have only one pair of parallel sides? 2. What properties do squares and rectangles share? 3. Which quadrilaterals are not part of the parallelogram family? c Answers may look like: 1. In the Venn diagram, the section representing quadrilaterals with only one pair of parallel sides would be trapezium. Therefore, trapezium has only one pair of parallel sides. 2. In the Venn diagram, both squares and rectangles have all angles equal to 90 degrees. 3. In the Venn diagram, the sections representing quadrilaterals that are not part of the parallelogram family would include trapezium and kite.

Chapter 15 review 1 C 2 a Impossible

b Likely

c Likely

d Unlikely

25 a 4

b 4

c 2

26 a

16.01 Types of data Understanding

3 a

4 a 0%

1 Categorical and numerical

b 50%

2 a ii

b i

c iii

Fluency

5 6 a 4 7 a Yes

b b Yes

c No

d Yes

e No 8 a O

9 a

10 a

11 12 68 13 2

17 a 58%

b 42%

h Numerical

i Categorical

j Categorical

k Numerical

l Categorical

4 a Discrete

b Discrete

c Discrete

d Continuous

e Continuous

f Continuous

g Discrete

h Continuous

i Continuous

j Continuous

k Discrete

l Discrete

m Continuous

n Discrete

o Discrete

p Continuous

e Continuous numerical f Discrete numerical g Categorical h Categorical

19 0.9990 or 1

i Continuous numerical b 10

j Discrete numerical

d No, because an experiment does not always go the way we expect. 21 a 157

b 33%

c 77%

22 a 70%

b 96%

c 86%

24 a 2

f Categorical

g Numerical

d Categorical

18 3%

b i

d Categorical

e Categorical

c Discrete numerical

16 a

23 a i 48

c Categorical

b Discrete numerical

15 32

20 a

b Categorical

5 a Continuous numerical

b No

14 a

3 a Numerical

ii 21

k Continuous numerical l Categorical Reasoning 6 a Continuous: the amount of liquid is measured b Discrete: the number of cans is counted and must be a whole number

c Discrete: the number of views is counted and must be a whole number

b 6

d Continuous: the amount of time is measured

Answers mathspace.co

879

7 a Continuous numerical: the amount of time is a number that is measured b Categorical: the types of pets are words, not numbers

8 a 107 people

b 50 people

c 42 people

9 a 25 people

b 11 people

c 32 people

10 a

Number of bedrooms Number of students

c Discrete numerical: the number of pets is counted and must be a whole number d Continuous numerical: the weight is a number that is measured 8 The question could say: “How many students scored higher than 70 in your most recent science test?” If it were written like this, the answer to the question would be a number.

b 45.5 km

11 a Categorical b Comedy 12 a Discrete numerical

c 20 people

b 2 people

c 9 hours

d 9 hours

13 a Categorical

b 6 cats

c 30 pets

d One dog

16.02 Frequency tables

1 a Mode

b Frequency b True

c False

d True

Fluency b 2, 5

c 23

4 7 5 a

Sport

Number of friends

Football

Rugby

Netball

Swimming

Cricket

b 15 friends

c Rugby

6 a 36 classmates

b 1 hour

c 30 classmates 7 a 149.9 cents

880

1 c 4 bedrooms

Animal

Number of friends

Cat

Dog

Fish

12 To find the total number of apples sold, you would multiply each number of apples by its corresponding frequency, and then add up the results. The calculation would be (10 × 5) + (15 × 8) + (20 × 12) + (25 × 6) = 560. This means that the total number of apples sold by the vendor in a week was 560. 13 a

Understanding

3 a 13

Reasoning

10 a Continuous data

2 a False

b 12 students

9 The question could say: “What is the brand of the shoes you are wearing?” If it were written like this, the answer to the question would be a word. Problem-solving

1 2

d 6 classmates b 17 days

Mathspace WA Curriculum – Year 7 mathspace.co

Colour

Frequency

Red

Blue

Green

Yellow

Purple

b The favourite colour among the students in the class is blue, with a frequency of 9 as it has the largest number. 14 Add all the frequency for vowels A, E, and I, 22 + 28 + 20 = 70. Next, subtract 70 from the total vowel letters on a piece of text (320 − 70 = 250) to determine the remaining vowels for O and U. Since O + U = 250 and O = 1.5U, we can determine the value of U by 1.5U + U = 250. Simplifying the equation, we have U = 100 and O = 150.

7 a • Red Rockets

Problem-solving 15 a

Hair colour

Frequency

Brown

Blonde

Black

Red

b 44 students

b If the Blue Blazers’ first game score changes from 70 to 90, their mean increases, their median increases, and their mode remains unchanged (no mode).

c 4.5% d

Sick days

Frequency

b 2 students 18 a

• Blue Blazers Mean: 82.4, Median: 85, Mode: None

16 a 15 trains b 2 trains c 17 a

Mean: 84, Median: 85, Mode: None

Mean: 60, Median: 60, Mode: None • Sarah Mean: 57, Median: 60, Mode: 60 b The mean, median, and mode all decrease for both sales representatives, with Sarah’s mode also changing to 45 and 60.

c 4 days

Weight in kilograms

Frequency

2 0

10 a 11

11 12

• Café Delight Mean: 8.6, Median: 9, Mode: 9

c $9

1 It measures the average of the values. 2 It measures the middle value of the set of values. b False

b 2

c 9

12 a i 14

ii 18

b i 19

ii 18

13 a 10 calls

Understanding

c True

d False

e False

b i 4 minutes

ii 3 minutes

iii 2 minutes

iv 11 minutes

14 a 9 tests b i 86

ii 87

iii 87

iv 10

15 Set A 16 Set D Reasoning

Fluency b 9.25

c 6

d 8.75

f 13

g 12.6

h 12.2

6 a 5

b 17.5

c 12.5

d 5

e 5

f 12

g 4.7

h 33.2

e 12.6

b • Café Delight Mean: 8.4, Median: 8, Mode: 8

16.03 Summarise data

4 a 6.5

9 a Bistro Bliss could have a slightly higher mean rating.

b 2 passengers

3 a False

8 a • Jacob

5 92

17 a Sometimes. If the data set has an odd number of values, the median is the middle value. If the data set has an even number of values, the median is between the two middle values and may not be in the data set. b Never. The mean will always be less than or equal to the maximum value in the set.

Answers mathspace.co

881

c Always. The range represents the difference between the largest and smallest values, and the difference tells us how far apart the values are. 18 The range would not change. 19 a The difference between the most points scored last season and the lowest points scored last season was 43 points. b The range shows that there is a big difference between the most points scored and the lowest points scored during the season. This means the team is not consistent with the number of goals they score, since sometimes they score a lot and other times they score less. If the coach can figure out why they didn’t score as many points in the lower-scoring games, the coach may be able to help them play better and score more points more often. 20 a These are the means of each season:

25 56 26 85 27 4, 5, 6, 7, 8 28 Example answers: • 9, 11, 13, 15 • 1, 12, 12, 20 • 2, 10, 14, 22 29 Example answer: 13, 14, 14, 16

16.04 Outliers Understanding 1 An outlier is a data point that varies significantly from the body of the data. An outlier will be a value that is either significantly larger or smaller than other observations. 2 a The mode does not change. b The range always decreases.

• Season 1 is 8.1

3 a Larger

• Season 2 is 8.45

Fluency

• Season 3 is 8.15 Season 2 has the greatest mean rating. b These are the medians of each season: • Season 1 is 8.10 • Season 2 is 8.40 • Season 3 is 9.05 Season 3 has the greatest median rating. c The answer to part (a) would change but the answer to part (b) would not. With a rating of 6.3 for Episode 6, the mean of Season 3 would be about 8.48, which is the highest mean of the three seasons. With a rating of 6.3 for Episode 6, the median would still be 9.05, which is still the highest median of the three seasons. Problem-solving

d Smaller

4 a 1

b 99

c 275

d 5.1

5 a 18 kg

b 14 kg

c 18 kg

d The mode are the same before and after removing the outlier. 6 a 18 kg

22 12th score 23 17th and 18th scores 24 15 goals

Mathspace WA Curriculum – Year 7 mathspace.co

b 13.3 kg

c Lower

7 a Increase

b Decrease

c Decrease

d Increase

8 a Yes

b No

9 a i

c Yes

Mean:

Median:

Mode:

Range:

102

Mean:

d No

ii 126 iii

21 11

882

b Smaller c Larger

Median:

Mode:

Range:

iv The mean, median and range of the data decreased, but the mode stayed the same.

b i

Mean:

1.92

Median:

1.7

Mode:

0.9

Range:

3.8

Mean:

1.61

Median:

1.5

Mode:

0.9

Range:

1.9

ii 4.7 iii

iv The mean, median and range decreased, but the mode stayed the same. c i

Mean:

4540

Median:

4750

Mode:

4700

Range:

3800

ii 1500 iii

Mean:

4877.8

Median:

4800

Mode:

4700

Range:

800

iv The mean and median increased, the range decreased, but the mode stayed the same. 10 a 12.5 kg c 13.875 kg

b 14.2 kg d 14.25 kg

Reasoning 11 The mean will be higher, but the median will stay the same. 12 a 15

b 13.29

c The median because it is not affected by the outlier. 13 She should use the mean since it is the higher measure of average because it is affected by the outlier. 14 Median, because it is the least affected by the outlier. 15 a Example answer: The mode. Since most people get paid $58 000, the owner might think it is the most representative of the company’s annual pay rate.

b Example answer: The mean. If Ted wants to get a raise, the mean will show the largest difference between the average pay rate and his pay rate. c Example answer: The median. A mathematician would recognise that $400 000 is an outlier and know that outliers affect the mean but do not always affect the median. Problem-solving 16 a 280

b A

17 a $462 000

b $527 000

d $929 000

e $446 600

c Median

18 x = 6 or x = 9 19 a The mean will be lower since the outlier is larger than the rest of the values. b The median will be unchanged since the middle of a large group of data points is mostly unchanged when extreme values are removed. c The mode of the data set is unchanged. d The range will decrease since the largest data value is being removed.

16.05 Dot plots and column graphs Understanding 1 By counting the number of dots above a particular category 2 The mode is the highest column of dots on a dot plot or the highest column on a column graph. 3 Column graph Fluency 4 a 4

b i 15

9 10 11 12 13 14 15 16 17 18 19

ii 15

iii 15

c 8 people 5 a The dots represent how many times she worked a particular number of shifts per week. b 3 weeks

c 6

d The distribution of the number of shifts per week is approximately symmetric.

Answers mathspace.co

883

6 a False

b True

7 a 2009 8

c True

d True

b 2006

Sport

12 a The distribution of ratings given by the judges to the winning contestant is positively skewed. b 7 judges

Number of People

Cricket

Basketball

Soccer

Tennis

Netball

13 a 1 goal

b 2 goals

14 a 15 people

b 16 names

c 10 names

d 6 names

b Underlook and Elf Fortress c 32 students 16 a Rating 4 d

Number of websites visited daily

Frequency

b The clusters in the number of websites visited daily are 2 and 4 websites, where the frequencies are relatively higher. c Yes, there is a gap in the distribution between the values 0 and 2, as there are no individuals who visited exactly 1 website daily. b 23 people

884

Mathspace WA Curriculum – Year 7 mathspace.co

18 Marbles

12 10 8 6 4 2 Black

18 a

Blue Yellow White Colour

Red

Sydney Opera House visitors 11

c 3 people

d The distribution of band members across different sections is negatively skewed, with the highest number of members in Strings and the lowest in Percussion.

14 Number of marbles

d Mean: 41.05, Median: 40

c 4 people

Frequency

Number of people (in millions)

c The distribution will become less skewed and more symmetric, as the outlier was pulling the mean towards the higher end of the scale. The median will remain unchanged, as it is not affected by the outlier.

b 18 ratings

Hotel Rating

b Mean: 49, Median: 40

11 a Percussion

d 10 judges

15 a Bash Sisters

9 a The scores have higher-frequency columns on the left (the lower scores), so the distribution is positively skewed. There appears to be an outlier as there is one value far to the right of the others. The data set also shows a large gap between 75 and 200, and clustering around 40.

10 a

c 16 judges

2015 2016 2017 2018 2019 Year

b Yes

19 a

b Yes, it would be reasonable. Dundalli’s points

Monthly attendance

Number of students

are actually 70 ÷ 30 =

Nemarluk’s points, so Dundalli scored more than twice as many points as Nemarluk. However, it would sound strange to say “Dundalli scored twice and a third more than Nemarluk, ” so it is reasonable to round down.

35 30 25 20 15 10 5 0

more than

Problem-solving August September October November December Month

b November

c October

20 a 48 aircraft

d 4

24 a 1.5 hours 25 a 59.5 seconds

d 25% aircrafts

b Mr. Lee’s class

c Mr. Lee’s class

b 10.6 mm

c 12.1 mm

c $130

d Mr. Navra’s class

26 a 12 games

b 12 points

c 105 points

Reasoning

d 8.75 points

21 The median would best represent the data as there is an outlier at 21 °C. The mean is not the best option, as outliers greatly affect the result. The mode is also not the best option, as there are two modes.

16.06 Stem and leaf plots

22 a No, Darius is not correct because he has not considered the frequency of the scores. If he had considered the frequency, he would find the median score is 48.

2 a 30

b Example answer:

1 a Stem, leaf (in order) b Leaf

b 6 is the stem and 5 is the leaf.

c 12

d 52

3 a True

b False

Swimming carnival results

Points

c Stem

c False

d True

Fluency

One method is to multiply the frequency of each column by the score for that column. After that, we need to add each of the products, then divide by the sum of the frequencies of all the columns. 23 a

Understanding

100 90 80 70 60 50 40 30 20 10

Age

Frequency

10–19

20–29

30–39

40–49

50–59

60–69

5 12 6

Truganini Nemarluk Dundalli Umbarra House

Stem 6 7 8 9 10

Ticket prices Leaf 0 4 6 7 3 5 6 6 2 4 4 5 1 4 6 7 4

7 7 9

Key: 62 = 62 dollars

Answers mathspace.co

885

7 22

8 44 9 a 19 concerts

b 56 tickets

c 46 tickets

10 a 19 people

b 11 years

c 53 years

d 20 years

e 35

11 a 25 km d

b 29 km

Stem 1 1 2 2 3 3 4 4

c 26.44 km

Leaf 1 6 1 5 0 5 0

2 7 3 5 1 7

3 8 4 5 1 8

Stem 9 9 10 10 11 11 12 12

3 6 0 5 2 5 0 5

3 8 3 6 4 8 1 7

9 4

Key: 12 = 12 kilometres Reasoning 12 To find the range, we find the smallest data value in the first row and subtract it from the largest data value in the last row of the stem and leaf plot.

One possible answer: The split stem and leaf plot shows that the readings are usually between 96 and 99. 17 a 8.7

b 29%

18 a 37 beats per minute b 179 beats per minute c 178 beats per minute d 164 beats per minute e 259 seconds longer 19 A = 9, B = 3, C = 6

16.07 Pie charts Understanding 1 a False

b True

c True

2 a No

b Yes

c No

13 The median or the mode. The mean would be affected by the outlier, so the median or mode would be a better representation of the centre of the data.

Fluency

14 a Mr. Ross

3 a Pizza

b Nuggets and noodles

4 a Snakis

b Wrinkles

b Since we are looking at the class as a whole, we need to consider all the data values. The only measure of centre that does this is the mean. 15 a There is no key provided. Each data point could be a whole number like 11 or decimals like 1.1. b The leaves in stem 2 are not in order. Problem-solving 16 a 127 units b 98 units c 30 readings d

886

Key: 112 = 112 units

2 8

3 7 0 5 3 6 1 6

Leaf 3 8 9 3 4 7 9 4 9 9 1 2

Mathspace WA Curriculum – Year 7 mathspace.co

c Hot Cheezos 5 a

b 50%

d Yes

d 224 high schoolers c

d 25%

6 a 400 roses

b 200 roses

c 120 roses

d 80 roses

7 a Palpatine 8 a i $60

b 1000 packages ii $120

iii $360

b $1800 9 a Sector D

b 26.3%

iv $300

Reasoning

Problem-solving

10 Example answer:

13 a

First, we could set up an equation to help us find the total number of fruits purchased: 0.4x = Number of grapes purchased where x represents the total number of fruits purchased. Next, we would solve for x by dividing both sides by 0.4. This would tell us the total number of fruits purchased. Finally, we would multiply the total number of fruits purchased by 27% to find the number of apples that were purchased.

Subject

Number of Percentage Angle size students

Maths

20% of 360 = 72°

English

25% of 360 = 90°

History

15% of 360 = 54°

Science

35% of 360 = 126°

Other

5% of 360 = 18°

b Favourite Subject

11 a Russia

Other

b China and France c 7 countries

d Example answer 1: Maths 20%

I agree with Elise. Since the number of nuclear weapons owned by other countries is so small, it is not possible to fit the data into one pie chart. If a column graph was used, all the data could be on the same graph.

Science 35%

English 25%

History 15%

Example answer 2: I disagree with Elise. Since the difference in the number of nuclear weapons owned by Russia and North Korea is so big, it would be difficult to choose a scale for the column sizes. The scale would need to be small enough to show the different heights between countries other than USA and Russia but big enough to fit USA and Russia on the same scale. Although the data cannot fit on the same pie chart, the two circles clearly show the differences in the number of nuclear weapons owned by all countries. 12 a The sector pieces are not the correct fraction of the circle. For example, the smallest percentage is the largest piece. Also, the sectors do not add to 100%. b B

c D

14 a

Result

Number of volunteers

Improvement

705

No change

240

Deterioration

135

Angle size

b Vaccine test results

Improvement 235°

No change 80° 45° Deterioration

Answers mathspace.co

887

15 a

2 If the scale is too small, small changes in the data can appear to be significant changes, which can be misleading. If the scale is too large, important changes or trends in the data may seem small or may get overlooked.

Month

Number of employees

November

December

January

February

4 a 32 °C

March

5 a 2300 visitors

Angle size

3 a Pie chart

b Line graph

c Dot plot

Fluency b 35 °C

d 33 °C

b 2650 visitors

c Month 5 and 6 b Holidays taken over summer

c 34 °C d No

Vegetables eaten

December

Number of vegetables

7 January

80° 100° 60° 40° November 80° February March

6 5 4 3 2 1

16 a

ManufacResponses turer Apple

2051

Samsung

967

Huawei

531

Angle size

Sun Mon Tue Wed Thur Fri

Sat

Day

10 9 8

451

Smartphone market share

Frequency

Other

7 6 5 4 3 2 Apple

Samsung

Huawei

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Other

8 a 2013

16.08 Line graphs Understanding 1 A line graph clearly shows how one value increases or decreases as another value changes.

888

Mathspace WA Curriculum – Year 7 mathspace.co

Score

b 2016

9 a Sunday

c 50

d 25

b Monday and Tuesday

c 7 °C warmer 10 a May c July to August

b October d February to March

e November to December

11 a 0.2 km, 0.9 km and 1.2 km

21 a They travelled 4 km closer to home.

b 0.6 km and 1 km c 0.2 km

d 0.5 m/s

b They travelled 1 km closer to home, but slower than before. c They travelled 4 km closer to home.

12 a 16 L

d They spent 1 minute not moving at all.

b The driver filled the tank. c 16 L

d 12 p.m.

13 a Increasing b $200 000 c June ‘13, September ‘13, and June ‘12 14 a Berlin

b 30 mm

c 23 mm

d London

e December 15 a $1.50

b December and June

c $2.75

d April and September

e $312.50

22 a The line graph only shows the data for the half of the year. Showing only some parts of the data makes the graph bias or misleading. b The y-axis starts at 0 which makes the line smaller showing small changes in each period. Problem-solving 23 a Apollo left home at 2:00 p.m., while Artemis left at 6:00 p.m. b They were the same distance from home sometime between 8:00 p.m. and 9:00 p.m. c Artemis

b 12:00 p.m.

d No

c 0.6 L

d 4 hours

24 a 20 km

e 1L

f 0L

16 a 1.6 L

g 2.4 L

25 a 4 km

b The first 4 minutes b 5 km

c 3 times d 9 km

e Between 4 and 5 hours

17 a 2.5 km

b 3:00 p.m.

c 3.5 km

d 4 hrs

Reasoning

26 a 2 hours into the journey

b 400 km

c The final 3 hours and the first 2 hours 27 a 4 km

b 3 km

c 9 hours into shift

18 D

d Pollux’s speed in his last two hours

Water

20 18 16 14 12 10 8 6 4 2 0

Chapter 16 review 1 a Discrete

b Nominal

c Continuous

d Ordinal

2 a 9 tests b i 90.8 1

3 a Yes

Time

ii 91

iii 92

iv 10

b No

c Yes

d No

4 a The number of days exercised in one week b 3 weeks

1080

5 a True

960

Water

840

b False

6 a White

720

c True

d False

b Green and red

c 44 vehicles

600 480

7 39

360 240 120 0

c 3

8 a 17 m

b 27 m

c 22.72 m

9 a Red

b Blue

c 112 fish

Time

Answers mathspace.co

889

10 a Basketball

b Water polo

11 a 1800 books

c 57 students

b 2200 books

c Month 5 and 6

d No

12 They should use the mean since it is the higher measure of average because it is affected by the outlier. 13 The median or the mode. The mean would be affected by the outlier, so the median or mode would be a better representation of the centre of the data.

c $45

17 F rom the stem and leaf plot, we can see that the data is fairly evenly distributed across the tens digits of 2, 3, and 4. There is a concentration of data in the 30s, indicating that most students spend between 30 and 40 minutes on homework each day. The distribution does not show significant skewness, clusters, or gaps, suggesting a relatively consistent amount of time spent on homework among students. Stem 2 3 4

5 0 0

7 1 2

8 2 5

Leaf 9 3 4

Key: 25 = 25 18 a John left home at 2:00 p.m., while Jane left at 3:00 p.m. b They were the same distance from home sometime between 3:00 p.m. and 4:00 p.m. c Jane

d No

19 a 3 km b 1.5 km c 4.5 hours into the shift d Ava’s speed in her last three hours. 20 a 21 b 21 c Answers can vary.

890

1 a 60

b 1

c 24

d 7

2 a Multiply by 60

b Dvide by 60

c Divide by 24

d Divide by 60

e Multiply by 3600

f M ultiply by 24 then multiply by 60

3 a a.m.

b p.m.

4 a 216 hours

b A

16 a 2 books

Understanding

c a.m.

d p.m.

Fluency

14 The score must be larger than 92. 15 a 100

17.01 Elapsed time

Mathspace WA Curriculum – Year 7 mathspace.co

b 660 seconds

c 2160 minutes

d 86 400 seconds

e 4 minutes

f 6 hours

g 13 minutes

h 13 hours

5 2 hours and 30 minutes 6 3 minutes 7 45 minutes 8 38 minutes 9 8760 hours 10 a 10:00 a.m.

b 12:20 a.m.

c 10:00 a.m.

d 7:24 p.m.

e 12:05 p.m.

f 4:10 p.m.

g 1:00 p.m.

h 10:13 p.m.

11 16 minutes 12 4 hours 13 4 hours 14 8 hours 15 3:00 p.m. 16 3:10 p.m. Reasoning 17 a The answer cannot be correct because time is represented in hours and minutes, not in decimal form. Adding the minutes together is correct, but the result should be converted back to hours and minutes. b Add the numbers of minutes together. If the answer comes to 60 or more, add one to the hours and subtract 60 from the minutes.

18 Yes. 365 ÷ 7 = 52.14 which rounds to 52, 366 ÷ 7 = 52.29 which also rounds to 52. 19 a 3 hours and 35 minutes

7 a 15:35 on Friday

b 22:10 on Wednesday

c 18:23 on Tuesday

d 19:33 on Sunday

e 21:57 on Friday

f 11:19 on Friday

b 4 hours and 0 minutes

8 a 23:45

b 13:20

c 22:30

c 4 hours and 10 minutes

d 11:15

e 08:10

f 15:50

d a + c hours and b + d minutes e a + c + 1 hours and 0 minutes f a + c + 1 hours and b + d − 60 minutes Problem-solving 20 a 3285 days c 4 730 400 minutes

9 a 20:10

b 8:10 p.m.

10 a 16:47

b 4:47 p.m.

11 a 07:25 or 19:25

b 02:50 or 14:50

c 08:11 b 78 840 hours

12 6 hours and 42 minutes

d 283 824 000 seconds

13 14:50

21 87 weeks

d 13:18

14 15:40 15 17:20

22 a 36.5 hours

b 2007.5 hours

16 21:20

23 9 years and 5 months

17 04:29

24 670 Martian days

18 15:12

25 a 144 minutes

b 86.4 seconds

c 0.864 seconds

Reasoning 19 a He is correct for times in the afternoon, but not for times in the morning when the hours are the same.

17.02 12 and 24-hour time Understanding 1 a.m. is used for times from 12 midnight until 11:59 in the morning, p.m. is used for times from 12 midday until 11:59 in the evening. 2 a Add 12 to the given time b Subtract 12 from the given time 3 As there are 24 hours in a day, the hours can be numbered from 0 to 23 when using 24-hour time. With 12-hour time, each number must be used twice, which is why a.m. and p.m. are used to distinguish between morning and afternoon.

b For example, 3:15 p.m. is 15:15 in 24-hour time, but 9:15 a.m. is 09:15 not 21:15. c Add 12 to the number of hours for times between 1:00 and 11:00 p.m. subtract 12 for times that are 12:00 a.m., don’t change the hours for 12:00 p.m. 20 12:25 a.m., 00:30, 12:25 p.m., 13:25, 19:30, 8:25 p.m. Problem-solving 21 9 hours and 20 minutes 22 14 hours and 20 minutes

Fluency 4 a 10:30

b 21:15

c 04:12

d 08:40

e 19:25

f 04:31

g 00:47

h 18:13

5 a 5:30 a.m.

b 6:30 p.m.

c 12:45 a.m.

d 1:59 p.m.

6 a 09:48 a.m.

b 12:28 p.m.

c 10:47 p.m.

d 12:56 a.m.

23 a 04:38

b 20:54

24 a 4:11 a.m.

b 4:38 p.m.

17.03 Australian time zones Understanding 1 Australian Western Standard Time (AWST) 2 UTC + 10.5

Answers mathspace.co

891

3 a Yes

b No

c Yes

13 a 12:25 p.m.

d No

e No

f Yes

b 9:55 a.m. c 5 hours and 22 mins.

Fluency

d 1 hour and 26 mins.

4 Canberra

e 7 hours and 18 mins.

5 a 4:42 p.m. b

Western Australia (WA)

South Queensland Australia (SA) (QLD)

c 10:45 a.m.

Chapter 17 review

9:45

11:15

11:45

8:45

10:15

10:45

5:25

6:55

7:25

AWST

ACST

AEST

3:55

5:25

5:55

2 a 168 hours

5:11

6:41

7:11

c 2880 minutes

d 43 200 seconds

10:25

11:55

12:25

e 5 minutes

f 6 hours

5:47

7:17

7:47

g 15 minutes

h 10 hours

6 a 15:14 AEST c 1:49 a.m. AEST 7 a 15:35 AEDT c 10:45 a.m. AEDT 8 a 21:20

7:45

8:15

b 7:00 a.m. AEST d 10:50 p.m. AEST b 5:00 a.m. AEDT d 6:13 p.m. AEDT b 22:20

Reasoning 9 Melbourne uses daylight saving, so the time difference is 3 hours. Adding 5.5 hours to 9 p.m. gives us 2:30 a.m. on Friday, but when we remove the 3 hour time difference the local time in Perth is 11:30 p.m. Queensland does not use daylight saving, so the time difference is only 2 hours. 10 During Daylight Saving Time, Sydney is UTC + 11 and Perth is UTC + 8, creating a 3-hour difference. A flight leaving at 6:00 a.m. AEDT is equivalent to 3:00 a.m. AWST. Adding the 5-hour flight duration results in an arrival time of 8:00 a.m. AWST. 11 Since both Melbourne and Sydney are in the same time zone (AEDT) during Daylight Saving Time, the train's arrival time is simply 3 hours after departure, making it 1:00 a.m. the next day. Problem-solving 12 22 hours

892

b 10:21 p.m.

1 A 12-hour time divides the day into two periods - a.m. and p.m., each lasting 12 hours. 24-hour time, on the other hand, divides the day into 24 hours running from midnight to midnight. For example, 3 p.m. in 12-hour time would be 15:00 in 24-hour time.

6:15

14 a 3:50 p.m.

Mathspace WA Curriculum – Year 7 mathspace.co

b 900 seconds

3 3 hours and 0 minutes 4 2.5 hours 5 10:50 a.m. 6 11:00 p.m. 7 a 18:55

b 14:25

c 22:30

d 22:45

e 14:45

f 06:05

8 a 17:15 on Tuesday

b 19:45 on Thursday

c 14:55 on Monday

d 13:35 on Monday

9 15:15 10 17:20 11 04:25 12 a 121.67 hours

b 4866.67 hours

13 10 503 Jovian days 14 a 4380 days c 6 307 200 minutes

b 105 120 hours d 378 432 000 seconds

15 12:45 a.m., 02:15, 12:45 p.m., 16:30, 18:45, 10:30 p.m. 16 a No

b Yes

17 a 1:49 a.m. AEST c 18:43 AEST 18 a 20:15

c Yes

d Yes

b 10:50 p.m. AEST d 21:56 AEST b 21:15

“Go down deep enough into anything and you will find mathematics.” Dean Schlicter